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Study Guides > Prealgebra

Problem Set 10: Polynomials

Practice Makes Perfect

Identify Polynomials, Monomials, Binomials and Trinomials In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. 5x+25x+2 binomial z25z6{z}^{2}-5z - 6 a2+9a+18{a}^{2}+9a+18 trinomial 12p4-12{p}^{4} y38y2+2y16{y}^{3}-8{y}^{2}+2y - 16 polynomial 109x10 - 9x 23y223{y}^{2} monomial m4+4m3+6m2+4m+1{m}^{4}+4{m}^{3}+6{m}^{2}+4m+1 Determine the Degree of Polynomials In the following exercises, determine the degree of each polynomial. 8a52a3+18{a}^{5}-2{a}^{3}+1 5 5c3+11c2c85{c}^{3}+11{c}^{2}-c - 8 3x123x - 12 1 4y+174y+17 13-13 0 22-22 Add and Subtract Monomials In the following exercises, add or subtract the monomials. 6x2+9x2{\text{6x}}^{2}+9{x}^{2} 15x2 4y3+6y3{\text{4y}}^{3}+6{y}^{3} 12u+4u-12u+4u −8u 3m+9m-3m+9m 5a+7b5a+7b 5a + 7b 8y+6z8y+6z Add: 4a,3b,8a\text{}4a,-3b,-8a −4a −3b Add: 4x,3y,3x4x,3y,-3x 18x2x18x - 2x 16x 13a3a13a - 3a Subtract 5x6from12x65{x}^{6}\text{from}-12{x}^{6} −17x6 Subtract 2p4from7p42{p}^{4}\text{from}-7{p}^{4} Add and Subtract Polynomials In the following exercises, add or subtract the polynomials. (4y2+10y+3)+(8y26y+5)\left(4{y}^{2}+10y+3\right)+\left(8{y}^{2}-6y+5\right) 12y2 + 4y + 8 (7x29x+2)+(6x24x+3)\left(7{x}^{2}-9x+2\right)+\left(6{x}^{2}-4x+3\right) (x2+6x+8)+(4x2+11x9)\left({x}^{2}+6x+8\right)+\left(-4{x}^{2}+11x - 9\right) −3x2 + 17x − 1 (y2+9y+4)+(2y25y1)\left({y}^{2}+9y+4\right)+\left(-2{y}^{2}-5y - 1\right) (3a2+7)+(a27a18)\left(3{a}^{2}+7\right)+\left({a}^{2}-7a - 18\right) 4a2 − 7a − 11 (p25p11)+(3p2+9)\left({p}^{2}-5p - 11\right)+\left(3{p}^{2}+9\right) (6m29m3)(2m2+m5)\left(6{m}^{2}-9m - 3\right)-\left(2{m}^{2}+m - 5\right) 4m2 − 10m + 2 (3n24n+1)(4n2n2)\left(3{n}^{2}-4n+1\right)-\left(4{n}^{2}-n - 2\right) (z2+8z+9)(z23z+1)\left({z}^{2}+8z+9\right)-\left({z}^{2}-3z+1\right) 11z + 8 (z27z+5)(z28z+6)\left({z}^{2}-7z+5\right)-\left({z}^{2}-8z+6\right) (12s215s)(s9)\left(12{s}^{2}-15s\right)-\left(s - 9\right) 12s2 − 16s + 9 (10r220r)(r8)\left(10{r}^{2}-20r\right)-\left(r - 8\right) Find the sum of (2p38)\left(2{p}^{3}-8\right) and (p2+9p+18)\left({p}^{2}+9p+18\right) 2p3 + p2 + 9p + 10 Find the sum of (q2+4q+13)\left({q}^{2}+4q+13\right) and (7q33)\left(7{q}^{3}-3\right) Subtract (7x24x+2)\left(7{x}^{2}-4x+2\right) from (8x2x+6)\left(8{x}^{2}-x+6\right) x2 + 3x + 4 Subtract (5x2x+12)\left(5{x}^{2}-x+12\right) from (9x26x20)\left(9{x}^{2}-6x - 20\right) Find the difference of (w2+w42)\left({w}^{2}+w - 42\right) and (w210w+24)\left({w}^{2}-10w+24\right) 11w − 66 Find the difference of (z23z18)\left({z}^{2}-3z - 18\right) and (z2+5z20)\left({z}^{2}+5z - 20\right) Evaluate a Polynomial for a Given Value In the following exercises, evaluate each polynomial for the given value. Evaluate8y23y+2\text{Evaluate}8{y}^{2}-3y+2y=5y=5y=2y=-2y=0y=0 ⓐ 187 ⓑ 40 ⓒ 2 Evaluate5y2y7when:\text{Evaluate}5{y}^{2}-y - 7\text{when:}y=4y=-4y=1y=1y=0y=0 Evaluate436xwhen:\text{Evaluate}4 - 36x\text{when:}x=3x=3x=0x=0x=1x=-1 ⓐ −104 ⓑ 4 ⓒ 40 Evaluate1636x2when:\text{Evaluate}16 - 36{x}^{2}\text{when:}x=1x=-1x=0x=0x=2x=2 A window washer drops a squeegee from a platform 275275 feet high. The polynomial 16t2+275-16{t}^{2}+275 gives the height of the squeegee tt seconds after it was dropped. Find the height after t=4t=4 seconds. 19 feet A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p dollars each is given by the polynomial 5p2+350p-5{p}^{2}+350p. Find the revenue received when p=50p=50 dollars.

Everyday Math

Fuel Efficiency The fuel efficiency (in miles per gallon) of a bus going at a speed of xx miles per hour is given by the polynomial 1160x2+12x-\frac{1}{160}{x}^{2}+\frac{1}{2}x. Find the fuel efficiency when x=40mph.x=40\text{mph.} 10 mpg Stopping Distance The number of feet it takes for a car traveling at xx miles per hour to stop on dry, level concrete is given by the polynomial 0.06x2+1.1x0.06{x}^{2}+1.1x. Find the stopping distance when x=60mph.x=60\text{mph.}

Writing Exercises

Using your own words, explain the difference between a monomial, a binomial, and a trinomial. Answers will vary. Eloise thinks the sum 5x2+3x45{x}^{2}+3{x}^{4} is 8x68{x}^{6}. What is wrong with her reasoning?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Practice Makes Perfect

Simplify Expressions with Exponents In the following exercises, simplify each expression with exponents. 45{4}^{5} 1,024 103{10}^{3} (12)2{\left(\frac{1}{2}\right)}^{2} 14\frac{1}{4} (35)2{\left(\frac{3}{5}\right)}^{2} (0.2)3{\left(0.2\right)}^{3} 0.008 (0.4)3{\left(0.4\right)}^{3} (5)4{\left(-5\right)}^{4} 625 (3)5{\left(-3\right)}^{5} 54{-5}^{4} −625 35{-3}^{5} 104{-10}^{4} −10,000 26{-2}^{6} (23)3{\left(-\frac{2}{3}\right)}^{3} 827-\frac{8}{27} (14)4{\left(-\frac{1}{4}\right)}^{4} 0.52-{0.5}^{2} −.25 0.14-{0.1}^{4} Simplify Expressions Using the Product Property of Exponents In the following exercises, simplify each expression using the Product Property of Exponents. x3x6{x}^{3}\cdot {x}^{6} x9 m4m2{m}^{4}\cdot {m}^{2} aa4a\cdot {a}^{4} a5 y12y{y}^{12}\cdot y 3539{3}^{5}\cdot {3}^{9} 314 51056{5}^{10}\cdot {5}^{6} zz2z3z\cdot {z}^{2}\cdot {z}^{3} z6 aa3a5a\cdot {a}^{3}\cdot {a}^{5} xax2{x}^{a}\cdot {x}^{2} xa+2 ypy3{y}^{p}\cdot {y}^{3} yayb{y}^{a}\cdot {y}^{b} ya+b xpxq{x}^{p}\cdot {x}^{q} Simplify Expressions Using the Power Property of Exponents In the following exercises, simplify each expression using the Power Property of Exponents. (u4)2{\left({u}^{4}\right)}^{2} u8 (x2)7{\left({x}^{2}\right)}^{7} (y5)4{\left({y}^{5}\right)}^{4} y20 (a3)2{\left({a}^{3}\right)}^{2} (102)6{\left({10}^{2}\right)}^{6} 1012 (28)3{\left({2}^{8}\right)}^{3} (x15)6{\left({x}^{15}\right)}^{6} x90 (y12)8{\left({y}^{12}\right)}^{8} (x2)y{\left({x}^{2}\right)}^{y} x2y (y3)x{\left({y}^{3}\right)}^{x} (5x)y{\left({5}^{x}\right)}^{y} 5xy (7a)b{\left({7}^{a}\right)}^{b} Simplify Expressions Using the Product to a Power Property In the following exercises, simplify each expression using the Product to a Power Property. (5a)2{\left(5a\right)}^{2} 25a2 (7x)2{\left(7x\right)}^{2} (6m)3{\left(-6m\right)}^{3} −216m3 (9n)3{\left(-9n\right)}^{3} (4rs)2{\left(4rs\right)}^{2} 16r2s2 (5ab)3{\left(5ab\right)}^{3} (4xyz)4{\left(4xyz\right)}^{4} 256x4y4z4 (5abc)3{\left(-5abc\right)}^{3} Simplify Expressions by Applying Several Properties In the following exercises, simplify each expression. (x2)4(x3)2{\left({x}^{2}\right)}^{4}\cdot {\left({x}^{3}\right)}^{2} x14 (y4)3(y5)2{\left({y}^{4}\right)}^{3}\cdot {\left({y}^{5}\right)}^{2} (a2)6(a3)8{\left({a}^{2}\right)}^{6}\cdot {\left({a}^{3}\right)}^{8} a36 (b7)5(b2)6{\left({b}^{7}\right)}^{5}\cdot {\left({b}^{2}\right)}^{6} (3x)2(5x){\left(3x\right)}^{2}\left(5x\right) 45x3 (2y)3(6y){\left(2y\right)}^{3}\left(6y\right) (5a)2(2a)3{\left(5a\right)}^{2}{\left(2a\right)}^{3} 200a5 (4b)2(3b)3{\left(4b\right)}^{2}{\left(3b\right)}^{3} (2m6)3{\left(2{m}^{6}\right)}^{3} 8m18 (3y2)4{\left(3{y}^{2}\right)}^{4} (10x2y)3{\left(10{x}^{2}y\right)}^{3} 1,000x6y3 (2mn4)5{\left(2m{n}^{4}\right)}^{5} (2a3b2)4{\left(-2{a}^{3}{b}^{2}\right)}^{4} 16a12b8 (10u2v4)3{\left(-10{u}^{2}{v}^{4}\right)}^{3} (23x2y)3{\left(\frac{2}{3}{x}^{2}y\right)}^{3} 827x6y3\frac{8}{27}{x}^{6}{y}^{3} (79pq4)2{\left(\frac{7}{9}p{q}^{4}\right)}^{2} (8a3)2(2a)4{\left(8{a}^{3}\right)}^{2}{\left(2a\right)}^{4} 1,024a10 (5r2)3(3r)2{\left(5{r}^{2}\right)}^{3}{\left(3r\right)}^{2} (10p4)3(5p6)2{\left(10{p}^{4}\right)}^{3}{\left(5{p}^{6}\right)}^{2} 25,000p24 (4x3)3(2x5)4{\left(4{x}^{3}\right)}^{3}{\left(2{x}^{5}\right)}^{4} (12x2y3)4(4x5y3)2{\left(\frac{1}{2}{x}^{2}{y}^{3}\right)}^{4}{\left(4{x}^{5}{y}^{3}\right)}^{2} x18y18 (13m3n2)4(9m8n3)2{\left(\frac{1}{3}{m}^{3}{n}^{2}\right)}^{4}{\left(9{m}^{8}{n}^{3}\right)}^{2} (3m2n)2(2mn5)4{\left(3{m}^{2}n\right)}^{2}{\left(2m{n}^{5}\right)}^{4} 144m8n22 (2pq4)3(5p6q)2{\left(2p{q}^{4}\right)}^{3}{\left(5{p}^{6}q\right)}^{2} Multiply Monomials In the following exercises, multiply the following monomials. (12x2)(5x4)\left(12{x}^{2}\right)\left(-5{x}^{4}\right) −60x6 (10y3)(7y2)\left(-10{y}^{3}\right)\left(7{y}^{2}\right) (8u6)(9u)\left(-8{u}^{6}\right)\left(-9u\right) 72u7 (6c4)(12c)\left(-6{c}^{4}\right)\left(-12c\right) (15r8)(20r3)\left(\frac{1}{5}{r}^{8}\right)\left(20{r}^{3}\right) 4r11 (14a5)(36a2)\left(\frac{1}{4}{a}^{5}\right)\left(36{a}^{2}\right) (4a3b)(9a2b6)\left(4{a}^{3}b\right)\left(9{a}^{2}{b}^{6}\right) 36a5b7 (6m4n3)(7mn5)\left(6{m}^{4}{n}^{3}\right)\left(7m{n}^{5}\right) (47xy2)(14xy3)\left(\frac{4}{7}x{y}^{2}\right)\left(14x{y}^{3}\right) 8x2y5 (58u3v)(24u5v)\left(\frac{5}{8}{u}^{3}v\right)\left(24{u}^{5}v\right) (23x2y)(34xy2)\left(\frac{2}{3}{x}^{2}y\right)\left(\frac{3}{4}x{y}^{2}\right) 12x3y3\frac{1}{2}{x}^{3}{y}^{3} (35m3n2)(59m2n3)\left(\frac{3}{5}{m}^{3}{n}^{2}\right)\left(\frac{5}{9}{m}^{2}{n}^{3}\right)

Everyday Math

Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is 62{6}^{2}, on the third round is 63{6}^{3}, as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.
Round Number of people
11 66
22 62{6}^{2}
33 63{6}^{3}
\dots \dots
88 ??
1,679,616 Salary Raul’s boss gives him a \text{5%} raise every year on his birthday. This means that each year, Raul’s salary is 1.051.05 times his last year’s salary. If his original salary was \text{$40,000} , his salary after 11 year was \text{$40,000}\left(1.05\right), after 22 years was \text{$40,000}{\left(1.05\right)}^{2}, after 33 years was \text{$40,000}{\left(1.05\right)}^{3}, as shown in the table below. What will Raul’s salary be after 1010 years? Simplify the expression, to show Raul’s salary in dollars.
Year Salary
11 \text{$40,000}\left(1.05\right)
22 \text{$40,000}{\left(1.05\right)}^{2}
33 \text{$40,000}{\left(1.05\right)}^{3}
\dots \dots
1010 ??

Writing Exercises

Use the Product Property for Exponents to explain why xx=x2x\cdot x={x}^{2}. Answers will vary. Explain why 53=(5)3{-5}^{3}={\left(-5\right)}^{3} but 54(5)4{-5}^{4}\ne {\left(-5\right)}^{4}. Jorge thinks (12)2{\left(\frac{1}{2}\right)}^{2} is 11. What is wrong with his reasoning? Answers will vary. Explain why x3x5{x}^{3}\cdot {x}^{5} is x8{x}^{8}, and not x15{x}^{15}.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ After reviewing this checklist, what will you do to become confident for all objectives?  

Practice Makes Perfect

Multiply a Polynomial by a Monomial In the following exercises, multiply. 4(x+10)4\left(x+10\right) 4x + 40 6(y+8)6\left(y+8\right) 15(r24)15\left(r - 24\right) 15r − 360 12(v30)12\left(v - 30\right) 3(m+11)-3\left(m+11\right) −3m − 33 4(p+15)-4\left(p+15\right) 8(z5)-8\left(z - 5\right) −8z + 40 3(x9)-3\left(x - 9\right) u(u+5)u\left(u+5\right) u2 + 5u q(q+7)q\left(q+7\right) n(n23n)n\left({n}^{2}-3n\right) n3 − 3n2 s(s26s)s\left({s}^{2}-6s\right) 12x(x10)12x\left(x - 10\right) 12x2 − 120x 9m(m11)9m\left(m - 11\right) 9a(3a+5)-9a\left(3a+5\right) −27a2 − 45a 4p(2p+7)-4p\left(2p+7\right) 6x(4x+y)6x\left(4x+y\right) 24x2 + 6xy 5a(9a+b)5a\left(9a+b\right) 5p(11p5q)5p\left(11p - 5q\right) 55p2 − 25pq 12u(3u4v)12u\left(3u - 4v\right) 3(v2+10v+25)3\left({v}^{2}+10v+25\right) 3v2 + 30v + 75 6(x2+8x+16)6\left({x}^{2}+8x+16\right) 2n(4n24n+1)2n\left(4{n}^{2}-4n+1\right) 8n3 − 8n2 + 2n 3r(2r26r+2)3r\left(2{r}^{2}-6r+2\right) 8y(y2+2y15)-8y\left({y}^{2}+2y - 15\right) −8y3 − 16y2 + 120y 5m(m2+3m18)-5m\left({m}^{2}+3m - 18\right) 5q3(q22q+6)5{q}^{3}\left({q}^{2}-2q+6\right) 5q5 − 10q4 + 30q3 9r3(r23r+5)9{r}^{3}\left({r}^{2}-3r+5\right) 4z2(3z2+12z1)-4{z}^{2}\left(3{z}^{2}+12z - 1\right) −12z4 − 48z3 + 4z2 3x2(7x2+10x1)-3{x}^{2}\left(7{x}^{2}+10x - 1\right) (2y9)y\left(2y - 9\right)y 2y2 − 9y (8b1)b\left(8b - 1\right)b (w6)8\left(w - 6\right)\cdot 8 8w − 48 (k4)5\left(k - 4\right)\cdot 5 Multiply a Binomial by a Binomial In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method (x+4)(x+6)\left(x+4\right)\left(x+6\right) x2 + 10x + 24 (u+8)(u+2)\left(u+8\right)\left(u+2\right) (n+12)(n3)\left(n+12\right)\left(n - 3\right) n2 + 9n − 36 (y+3)(y9)\left(y+3\right)\left(y - 9\right) In the following exercises, multiply the following binomials. Use any method. (y+8)(y+3)\left(y+8\right)\left(y+3\right) y2 + 11y + 24 (x+5)(x+9)\left(x+5\right)\left(x+9\right) (a+6)(a+16)\left(a+6\right)\left(a+16\right) a2 + 22a + 96 (q+8)(q+12)\left(q+8\right)\left(q+12\right) (u5)(u9)\left(u - 5\right)\left(u - 9\right) u2 − 14u + 45 (r6)(r2)\left(r - 6\right)\left(r - 2\right) (z10)(z22)\left(z - 10\right)\left(z - 22\right) z2 − 32z + 220 (b5)(b24)\left(b - 5\right)\left(b - 24\right) (x4)(x+7)\left(x - 4\right)\left(x+7\right) x2 + 3x − 28 (s3)(s+8)\left(s - 3\right)\left(s+8\right) (v+12)(v5)\left(v+12\right)\left(v - 5\right) v2 + 7v − 60 (d+15)(d4)\left(d+15\right)\left(d - 4\right) (6n+5)(n+1)\left(6n+5\right)\left(n+1\right) 6n2 + 11n + 5 (7y+1)(y+3)\left(7y+1\right)\left(y+3\right) (2m9)(10m+1)\left(2m - 9\right)\left(10m+1\right) 20m2 − 88m − 9 (5r4)(12r+1)\left(5r - 4\right)\left(12r+1\right) (4c1)(4c+1)\left(4c - 1\right)\left(4c+1\right) 16c2 − 1 (8n1)(8n+1)\left(8n - 1\right)\left(8n+1\right) (3u8)(5u14)\left(3u - 8\right)\left(5u - 14\right) 15u2 − 82u + 112 (2q5)(7q11)\left(2q - 5\right)\left(7q - 11\right) (a+b)(2a+3b)\left(a+b\right)\left(2a+3b\right) 2a2 + 5ab + 3b2 (r+s)(3r+2s)\left(r+s\right)\left(3r+2s\right) (5xy)(x4)\left(5x-y\right)\left(x - 4\right) 5x2 − 20xxy + 4y (4zy)(z6)\left(4z-y\right)\left(z - 6\right) Multiply a Trinomial by a Binomial In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method. (u+4)(u2+3u+2)\left(u+4\right)\left({u}^{2}+3u+2\right) u3 + 7u2 + 14u + 8 (x+5)(x2+8x+3)\left(x+5\right)\left({x}^{2}+8x+3\right) (a+10)(3a2+a5)\left(a+10\right)\left(3{a}^{2}+a - 5\right) 3a3 + 31a2 + 5a − 50 (n+8)(4n2+n7)\left(n+8\right)\left(4{n}^{2}+n - 7\right) In the following exercises, multiply. Use either method. (y6)(y210y+9)\left(y - 6\right)\left({y}^{2}-10y+9\right) y3 − 16y2 + 69y − 54 (k3)(k28k+7)\left(k - 3\right)\left({k}^{2}-8k+7\right) (2x+1)(x25x6)\left(2x+1\right)\left({x}^{2}-5x - 6\right) 2x3 − 9x2 − 17x − 6 (5v+1)(v26v10)\left(5v+1\right)\left({v}^{2}-6v - 10\right)

Everyday Math

Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 1313 times 1515. Think of 1313 as 10+310+3 and 1515 as 10+510+5.
  1. ⓐ Multiply (10+3)(10+5)\left(10+3\right)\left(10+5\right) by the FOIL method.
  2. ⓑ Multiply 131513\cdot 15 without using a calculator.
  3. ⓒ Which way is easier for you? Why?
  1. ⓐ 195
  2. ⓑ 195
  3. ⓐ Answers will vary.
Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 1818 times 1717. Think of 1818 as 20220 - 2 and 1717 as 20320 - 3.
  1. ⓐ Multiply (202)(203)\left(20 - 2\right)\left(20 - 3\right) by the FOIL method.
  2. ⓑ Multiply 181718\cdot 17 without using a calculator.
  3. ⓒ Which way is easier for you? Why?

Writing Exercises

Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why? Answers will vary. Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Practice Makes Perfect

Simplify Expressions Using the Quotient Property of Exponents In the following exercises, simplify. 4842\frac{{4}^{8}}{{4}^{2}} 46 31234\frac{{3}^{12}}{{3}^{4}} x12x3\frac{{x}^{12}}{{x}^{3}} x9 u9u3\frac{{u}^{9}}{{u}^{3}} r5r\frac{{r}^{5}}{r} r4 y4y\frac{{y}^{4}}{y} y4y20\frac{{y}^{4}}{{y}^{20}} 1y16\frac{1}{{y}^{16}} x10x30\frac{{x}^{10}}{{x}^{30}} 1031015\frac{{10}^{3}}{{10}^{15}} 11012\frac{1}{{10}^{12}} r2r8\frac{{r}^{2}}{{r}^{8}} aa9\frac{a}{{a}^{9}} 1a8\frac{1}{{a}^{8}} 225\frac{2}{{2}^{5}} Simplify Expressions with Zero Exponents In the following exercises, simplify. 50{5}^{0} 1 100{10}^{0} a0{a}^{0} 1 x0{x}^{0} 70-{7}^{0} −1 40-{4}^{0}(10p)0{\left(10p\right)}^{0}10p010{p}^{0} ⓐ 1 ⓑ 10 ⓐ (3a)0{\left(3a\right)}^{0}3a03{a}^{0}(27x5y)0{\left(-27{x}^{5}y\right)}^{0}27x5y0-27{x}^{5}{y}^{0} ⓐ 1 ⓑ −27x5(92y8z)0{\left(-92{y}^{8}z\right)}^{0}92y8z0-92{y}^{8}{z}^{0}150{15}^{0}151{15}^{1} ⓐ 1 ⓑ 15 ⓐ 60-{6}^{0}61-{6}^{1} 2x0+5y02\cdot {x}^{0}+5\cdot {y}^{0} 7 8m04n08\cdot {m}^{0}-4\cdot {n}^{0} Simplify Expressions Using the Quotient to a Power Property In the following exercises, simplify. (32)5{\left(\frac{3}{2}\right)}^{5} 24332\frac{243}{32} (45)3{\left(\frac{4}{5}\right)}^{3} (m6)3{\left(\frac{m}{6}\right)}^{3} m3216\frac{{m}^{3}}{216} (p2)5{\left(\frac{p}{2}\right)}^{5} (xy)10{\left(\frac{x}{y}\right)}^{10} x10y10\frac{{x}^{10}}{{y}^{10}} (ab)8{\left(\frac{a}{b}\right)}^{8} (a3b)2{\left(\frac{a}{3b}\right)}^{2} a29b2\frac{{a}^{2}}{9{b}^{2}} (2xy)4{\left(\frac{2x}{y}\right)}^{4} Simplify Expressions by Applying Several Properties In the following exercises, simplify. (x2)4x5\frac{{\left({x}^{2}\right)}^{4}}{{x}^{5}} x3 (y4)3y7\frac{{\left({y}^{4}\right)}^{3}}{{y}^{7}} (u3)4u10\frac{{\left({u}^{3}\right)}^{4}}{{u}^{10}} u2 (y2)5y6\frac{{\left({y}^{2}\right)}^{5}}{{y}^{6}} y8(y5)2\frac{{y}^{8}}{{\left({y}^{5}\right)}^{2}} 1y2\frac{1}{{y}^{2}} p11(p5)3\frac{{p}^{11}}{{\left({p}^{5}\right)}^{3}} r5r4r\frac{{r}^{5}}{{r}^{4}\cdot r} 1 a3a4a7\frac{{a}^{3}\cdot {a}^{4}}{{a}^{7}} (x2x8)3{\left(\frac{{x}^{2}}{{x}^{8}}\right)}^{3} 1x18\frac{1}{{x}^{18}} (uu10)2{\left(\frac{u}{{u}^{10}}\right)}^{2} (a4a6a3)2{\left(\frac{{a}^{4}\cdot {a}^{6}}{{a}^{3}}\right)}^{2} a14 (x3x8x4)3{\left(\frac{{x}^{3}\cdot {x}^{8}}{{x}^{4}}\right)}^{3} (y3)5(y4)3\frac{{\left({y}^{3}\right)}^{5}}{{\left({y}^{4}\right)}^{3}} y3 (z6)2(z2)4\frac{{\left({z}^{6}\right)}^{2}}{{\left({z}^{2}\right)}^{4}} (x3)6(x4)7\frac{{\left({x}^{3}\right)}^{6}}{{\left({x}^{4}\right)}^{7}} 1x10\frac{1}{{x}^{10}} (x4)8(x5)7\frac{{\left({x}^{4}\right)}^{8}}{{\left({x}^{5}\right)}^{7}} (2r35s)4{\left(\frac{2{r}^{3}}{5s}\right)}^{4} 16r12625s4\frac{16{r}^{12}}{625{s}^{4}} (3m24n)3{\left(\frac{3{m}^{2}}{4n}\right)}^{3} (3y2y5y15y8)0{\left(\frac{3{y}^{2}\cdot {y}^{5}}{{y}^{15}\cdot {y}^{8}}\right)}^{0} 1 (15z4z90.3z2)0{\left(\frac{15{z}^{4}\cdot {z}^{9}}{0.3{z}^{2}}\right)}^{0} (r2)5(r4)2(r3)7\frac{{\left({r}^{2}\right)}^{5}{\left({r}^{4}\right)}^{2}}{{\left({r}^{3}\right)}^{7}} 1r3\frac{1}{{r}^{3}} (p4)2(p3)5(p2)9\frac{{\left({p}^{4}\right)}^{2}{\left({p}^{3}\right)}^{5}}{{\left({p}^{2}\right)}^{9}} (3x4)3(2x3)2(6x5)2\frac{{\left(3{x}^{4}\right)}^{3}{\left(2{x}^{3}\right)}^{2}}{{\left(6{x}^{5}\right)}^{2}} 3x8 (2y3)4(3y4)2(6y3)2\frac{{\left(-2{y}^{3}\right)}^{4}{\left(3{y}^{4}\right)}^{2}}{{\left(-6{y}^{3}\right)}^{2}} Divide Monomials In the following exercises, divide the monomials. 48b8÷6b248{b}^{8}\div 6{b}^{2} 8b6 42a14÷6a242{a}^{14}\div 6{a}^{2} 36x3÷(2x9)36{x}^{3}\div \left(-2{x}^{9}\right) 18x6\frac{-18}{{x}^{6}} 20u8÷(4u6)20{u}^{8}\div \left(-4{u}^{6}\right) 18x39x2\frac{18{x}^{3}}{9{x}^{2}} 2x 36y94y7\frac{36{y}^{9}}{4{y}^{7}} 35x742x13\frac{-35{x}^{7}}{-42{x}^{13}} 56x6\frac{5}{6{x}^{6}} 18x527x9\frac{18{x}^{5}}{-27{x}^{9}} 18r5s3r3s9\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}} 6r2s8\frac{6{r}^{2}}{{s}^{8}} 24p7q6p2q5\frac{24{p}^{7}q}{6{p}^{2}{q}^{5}} 8mn1064mn4\frac{8m{n}^{10}}{64m{n}^{4}} n68\frac{{n}^{6}}{8} 10a4b50a2b6\frac{10{a}^{4}b}{50{a}^{2}{b}^{6}} 12x4y915x6y3\frac{-12{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}} 4y65x2-\frac{4{y}^{6}}{5{x}^{2}} 48x11y9z336x6y8z5\frac{48{x}^{11}{y}^{9}{z}^{3}}{36{x}^{6}{y}^{8}{z}^{5}} 64x5y9z748x7y12z6\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}} 4z3x2y3\frac{4z}{3{x}^{2}{y}^{3}} (10u2v)(4u3v6)5u9v2\frac{\left(10{u}^{2}v\right)\left(4{u}^{3}{v}^{6}\right)}{5{u}^{9}{v}^{2}} (6m2n)(5m4n3)3m10n2\frac{\left(6{m}^{2}n\right)\left(5{m}^{4}{n}^{3}\right)}{3{m}^{10}{n}^{2}} 10n2m4\frac{10{n}^{2}}{{m}^{4}} (6a4b3)(4ab5)(12a8b)(a3b)\frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{8}b\right)\left({a}^{3}b\right)} (4u5v4)(15u8v)(12u3v)(u6v)\frac{\left(4{u}^{5}{v}^{4}\right)\left(15{u}^{8}v\right)}{\left(12{u}^{3}v\right)\left({u}^{6}v\right)} 5u4v3

Mixed Practice

24a5+2a524{a}^{5}+2{a}^{5}24a52a524{a}^{5}-2{a}^{5}24a52a524{a}^{5}\cdot 2{a}^{5}24a5÷2a524{a}^{5}\div 2{a}^{5}15n10+3n1015{n}^{10}+3{n}^{10}15n103n1015{n}^{10}-3{n}^{10}15n103n1015{n}^{10}\cdot 3{n}^{10}15n10÷3n1015{n}^{10}\div 3{n}^{10}18n1018{n}^{10}12n1012{n}^{10}45n2045{n}^{20}55p4p6{p}^{4}\cdot {p}^{6}(p4)6{\left({p}^{4}\right)}^{6}q5q3{q}^{5}\cdot {q}^{3}(q5)3{\left({q}^{5}\right)}^{3}q8{q}^{8}q15{q}^{15}y3y\frac{{y}^{3}}{y}yy3\frac{y}{{y}^{3}}z6z5\frac{{z}^{6}}{{z}^{5}}z5z6\frac{{z}^{5}}{{z}^{6}}zz1z\frac{1}{z} (8x5)(9x)÷6x3\left(8{x}^{5}\right)\left(9x\right)\div 6{x}^{3} (4y5)(12y7)÷8y2\left(4{y}^{5}\right)\left(12{y}^{7}\right)\div 8{y}^{2} 6y66{y}^{6} 27a73a3+54a99a5\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}} 32c114c5+42c96c3\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}} 15c615{c}^{6} 32y58y260y105y7\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}} 48x66x435x97x7\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}} 3x23{x}^{2} 63r6s39r4s272r2s26s\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s} 56y4z57y3z345y2z25y\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y} yz2y{z}^{2}

Everyday Math

Memory One megabyte is approximately 106{10}^{6} bytes. One gigabyte is approximately 109{10}^{9} bytes. How many megabytes are in one gigabyte? Memory One megabyte is approximately 106{10}^{6} bytes. One terabyte is approximately 1012{10}^{12} bytes. How many megabytes are in one terabyte? 1,000,000

Writing Exercises

Vic thinks the quotient x20x4\frac{{x}^{20}}{{x}^{4}} simplifies to x5{x}^{5}. What is wrong with his reasoning? Mai simplifies the quotient y3y\frac{{y}^{3}}{y} by writing )y3)y=3\frac{{\overline{)y}}^{3}}{\overline{)y}}=3. What is wrong with her reasoning? Answers will vary. When Dimple simplified 30-{3}^{0} and (3)0{\left(-3\right)}^{0} she got the same answer. Explain how using the Order of Operations correctly gives different answers. Roxie thinks n0{n}^{0} simplifies to 00. What would you say to convince Roxie she is wrong? Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Practice Makes Perfect

Use the Definition of a Negative Exponent In the following exercises, simplify. 53{5}^{-3} 82{8}^{-2} 164\frac{1}{64} 34{3}^{-4} 25{2}^{-5} 132\frac{1}{32} 71{7}^{-1} 101{10}^{-1} 110\frac{1}{10} 23+22{2}^{-3}+{2}^{-2} 32+31{3}^{-2}+{3}^{-1} 49\frac{4}{9} 31+41{3}^{-1}+{4}^{-1} 101+21{10}^{-1}+{2}^{-1} 35\frac{3}{5} 100101+102{10}^{0}-{10}^{-1}+{10}^{-2} 2021+22{2}^{0}-{2}^{-1}+{2}^{-2} 34\frac{3}{4}(6)2{\left(-6\right)}^{-2}62-{6}^{-2}(8)2{\left(-8\right)}^{-2}82-{8}^{-2}164\frac{1}{64}164-\frac{1}{64}(10)4{\left(-10\right)}^{-4}104-{10}^{-4}(4)6{\left(-4\right)}^{-6}46-{4}^{-6}14096\frac{1}{4096}14096-\frac{1}{4096}5215\cdot {2}^{-1}(52)1{\left(5\cdot 2\right)}^{-1}103110\cdot {3}^{-1}(103)1{\left(10\cdot 3\right)}^{-1}103\frac{10}{3}130\frac{1}{30}41034\cdot {10}^{-3}(410)3{\left(4\cdot 10\right)}^{-3}3523\cdot {5}^{-2}(35)2{\left(3\cdot 5\right)}^{-2}325\frac{3}{25}1225\frac{1}{225} n4{n}^{-4} p3{p}^{-3} 1p3\frac{1}{{p}^{3}} c10{c}^{-10} m5{m}^{-5} 1m5\frac{1}{{m}^{5}}4x14{x}^{-1}(4x)1{\left(4x\right)}^{-1}(4x)1{\left(-4x\right)}^{-1}3q13{q}^{-1}(3q)1{\left(3q\right)}^{-1}(3q)1{\left(-3q\right)}^{-1}3q\frac{3}{q}13q\frac{1}{3q}13q-\frac{1}{3q}6m16{m}^{-1}(6m)1{\left(6m\right)}^{-1}(6m)1{\left(-6m\right)}^{-1}10k110{k}^{-1}(10k)1{\left(10k\right)}^{-1}(10k)1{\left(-10k\right)}^{-1}10k\frac{10}{k}110k\frac{1}{10k}110k-\frac{1}{10k} Simplify Expressions with Integer Exponents In the following exercises, simplify. p4p8{p}^{-4}\cdot {p}^{8} r2r5{r}^{-2}\cdot {r}^{5} r3 n10n2{n}^{-10}\cdot {n}^{2} q8q3{q}^{-8}\cdot {q}^{3} 1q5\frac{1}{{q}^{5}} k3k2{k}^{-3}\cdot {k}^{-2} z6z2{z}^{-6}\cdot {z}^{-2} 1z8\frac{1}{{z}^{8}} aa4a\cdot {a}^{-4} mm2m\cdot {m}^{-2} 1m\frac{1}{m} p5p2p4{p}^{5}\cdot {p}^{-2}\cdot {p}^{-4} x4x2x3{x}^{4}\cdot {x}^{-2}\cdot {x}^{-3} 1x\frac{1}{x} a3b3{a}^{3}{b}^{-3} u2v2{u}^{2}{v}^{-2} u2v2\frac{{u}^{2}}{{v}^{2}} (x5y1)(x10y3)\left({x}^{5}{y}^{-1}\right)\left({x}^{-10}{y}^{-3}\right) (a3b3)(a5b1)\left({a}^{3}{b}^{-3}\right)\left({a}^{-5}{b}^{-1}\right) 1a2b4\frac{1}{{a}^{2}{b}^{4}} (uv2)(u5v4)\left(u{v}^{-2}\right)\left({u}^{-5}{v}^{-4}\right) (pq4)(p6q3)\left(p{q}^{-4}\right)\left({p}^{-6}{q}^{-3}\right) 1p5q7\frac{1}{{p}^{5}{q}^{7}} (2r3s9)(6r4s5)\left(-2{r}^{-3}{s}^{9}\right)\left(6{r}^{4}{s}^{-5}\right) (3p5q8)(7p2q3)\left(-3{p}^{-5}{q}^{8}\right)\left(7{p}^{2}{q}^{-3}\right) 21q5p3-\frac{21{q}^{5}}{{p}^{3}} (6m8n5)(9m4n2)\left(-6{m}^{-8}{n}^{-5}\right)\left(-9{m}^{4}{n}^{2}\right) (8a5b4)(4a2b3)\left(-8{a}^{-5}{b}^{-4}\right)\left(-4{a}^{2}{b}^{3}\right) 32a3b\frac{32}{{a}^{3}b} (a3)3{\left({a}^{3}\right)}^{-3} (q10)10{\left({q}^{10}\right)}^{-10} 1q100\frac{1}{{q}^{100}} (n2)1{\left({n}^{2}\right)}^{-1} (x4)1{\left({x}^{4}\right)}^{-1} 1x4\frac{1}{{x}^{4}} (y5)4{\left({y}^{-5}\right)}^{4} (p3)2{\left({p}^{-3}\right)}^{2} 1y6\frac{1}{{y}^{6}} (q5)2{\left({q}^{-5}\right)}^{-2} (m2)3{\left({m}^{-2}\right)}^{-3} m6 (4y3)2{\left(4{y}^{-3}\right)}^{2} (3q5)2{\left(3{q}^{-5}\right)}^{2} 9q10\frac{9}{{q}^{10}} (10p2)5{\left(10{p}^{-2}\right)}^{-5} (2n3)6{\left(2{n}^{-3}\right)}^{-6} n1864\frac{{n}^{18}}{64} u9u2\frac{{u}^{9}}{{u}^{-2}} b5b3\frac{{b}^{5}}{{b}^{-3}} b8 x6x4\frac{{x}^{-6}}{{x}^{4}} m5m2\frac{{m}^{5}}{{m}^{-2}} m7 q3q12\frac{{q}^{3}}{{q}^{12}} r6r9\frac{{r}^{6}}{{r}^{9}} 1r3\frac{1}{{r}^{3}} n4n10\frac{{n}^{-4}}{{n}^{-10}} p3p6\frac{{p}^{-3}}{{p}^{-6}} p3 Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 45,000 280,000 2.8 × 105 8,750,000 1,290,000 1.29 × 106 0.036 0.041 4.1 × 10−2 0.00000924 0.0000103 1.03 × 10−5 The population of the United States on July 4, 2010 was almost 310,000,000310,000,000. The population of the world on July 4, 2010 was more than 6,850,000,0006,850,000,000. 6.85 × 109 The average width of a human hair is 0.00180.0018 centimeters. The probability of winning the 20102010 Megamillions lottery is about 0.00000000570.0000000057. 5.7 × 10−9 Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 4.1×1024.1\times {10}^{2} 8.3×1028.3\times {10}^{2} 830 5.5×1085.5\times {10}^{8} 1.6×10101.6\times {10}^{10} 16,000,000,000 3.5×1023.5\times {10}^{-2} 2.8×1022.8\times {10}^{-2} 0.028 1.93×1051.93\times {10}^{-5} 6.15×1086.15\times {10}^{-8} 0.0000000615 In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was 2×1042\times {10}^{4}. At the start of 2012, the US federal budget had a deficit of more than \text{$1.5}\times {10}^{13}. $15,000,000,000,000 The concentration of carbon dioxide in the atmosphere is 3.9×1043.9\times {10}^{-4}. The width of a proton is 1×1051\times {10}^{-5} of the width of an atom. 0.00001 Multiply and Divide Using Scientific Notation In the following exercises, multiply or divide and write your answer in decimal form. (2×105)(2×109)\left(2\times {10}^{5}\right)\left(2\times {10}^{-9}\right) (3×102)(1×105)\left(3\times {10}^{2}\right)\left(1\times {10}^{-5}\right) 0.003 (1.6×102)(5.2×106)\left(1.6\times {10}^{-2}\right)\left(5.2\times {10}^{-6}\right) (2.1×104)(3.5×102)\left(2.1\times {10}^{-4}\right)\left(3.5\times {10}^{-2}\right) 0.00000735 6×1043×102\frac{6\times {10}^{4}}{3\times {10}^{-2}} 8×1064×101\frac{8\times {10}^{6}}{4\times {10}^{-1}} 200,000 7×1021×108\frac{7\times {10}^{-2}}{1\times {10}^{-8}} 5×1031×1010\frac{5\times {10}^{-3}}{1\times {10}^{-10}} 50,000,000

Everyday Math

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by 1.51.5 trillion calories by the end of 2015.
  1. ⓐ Write 1.51.5 trillion in decimal notation.
  2. ⓑ Write 1.51.5 trillion in scientific notation.
Length of a year The difference between the calendar year and the astronomical year is 0.0001250.000125 day.
  1. ⓐ Write this number in scientific notation.
  2. ⓑ How many years does it take for the difference to become 1 day?
  1. ⓐ 1.25 × 10−4
  2. ⓐ 8,000
Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided 11 by 2,598,9602,598,960 and saw the answer 3.848×1073.848\times {10}^{-7}. Write the number in decimal notation. Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with 66 of her 5050 favorite photographs, she multiplied 50494847464550\cdot 49\cdot 48\cdot 47\cdot 46\cdot 45. Her calculator gave the answer 1.1441304×10101.1441304\times {10}^{10}. Write the number in decimal notation. 11,441,304,000

Writing Exercises

  1. ⓐ Explain the meaning of the exponent in the expression 23{2}^{3}.
  2. ⓑ Explain the meaning of the exponent in the expression 23{2}^{-3}
When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative? Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions In the following exercises, find the greatest common factor. 40,5640,56 45,7545,75 15 72,16272,162 150,275150,275 25 3x,123x,12 4y,284y,28 4 10a,5010a,50 5b,305b,30 5 16y,24y216y,24{y}^{2} 9x,15x29x,15{x}^{2} 3x 18m3,36m218{m}^{3},36{m}^{2} 12p4,48p312{p}^{4},48{p}^{3} 12p3 10x,25x2,15x310x,25{x}^{2},15{x}^{3} 18a,6a2,22a318a,6{a}^{2},22{a}^{3} 2a 24u,6u2,30u324u,6{u}^{2},30{u}^{3} 40y,10y2,90y340y,10{y}^{2},90{y}^{3} 10y 15a4,9a5,21a615{a}^{4},9{a}^{5},21{a}^{6} 35x3,10x4,5x535{x}^{3},10{x}^{4},5{x}^{5} 5x3 27y2,45y3,9y427{y}^{2},45{y}^{3},9{y}^{4} 14b2,35b3,63b414{b}^{2},35{b}^{3},63{b}^{4} 7b2 Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. 2x+82x+8 5y+155y+15 5(y + 3) 3a243a - 24 4b204b - 20 4(b − 5) 9y99y - 9 7x77x - 7 7(x − 1) 5m2+20m+355{m}^{2}+20m+35 3n2+21n+123{n}^{2}+21n+12 3(n2 + 7n + 4) 8p2+32p+488{p}^{2}+32p+48 6q2+30q+426{q}^{2}+30q+42 6(q2 + 5q + 7) 8q2+15q8{q}^{2}+15q 9c2+22c9{c}^{2}+22c c(9c + 22) 13k2+5k13{k}^{2}+5k 17x2+7x17{x}^{2}+7x x(17x + 7) 5c2+9c5{c}^{2}+9c 4q2+7q4{q}^{2}+7q q(4q + 7) 5p2+25p5{p}^{2}+25p 3r2+27r3{r}^{2}+27r 3r(r + 9) 24q212q24{q}^{2}-12q 30u210u30{u}^{2}-10u 10u(3u − 1) yz+4zyz+4z ab+8bab+8b b(a + 8) 60x6x360x - 6{x}^{3} 55y11y455y - 11{y}^{4} 11y(5 − y3) 48r412r348{r}^{4}-12{r}^{3} 45c315c245{c}^{3}-15{c}^{2} 15c2(3c − 1) 4a34ab24{a}^{3}-4a{b}^{2} 6c36cd26{c}^{3}-6c{d}^{2} 6c(c2d2) 30u3+80u230{u}^{3}+80{u}^{2} 48x3+72x248{x}^{3}+72{x}^{2} 24x2(2x + 3) 120y6+48y4120{y}^{6}+48{y}^{4} 144a6+90a3144{a}^{6}+90{a}^{3} 18a3(8a3 + 5) 4q2+24q+284{q}^{2}+24q+28 10y2+50y+4010{y}^{2}+50y+40 10(y2 + 5y + 4) 15z230z9015{z}^{2}-30z - 90 12u236u10812{u}^{2}-36u - 108 12(u2 − 3u − 9) 3a424a3+18a23{a}^{4}-24{a}^{3}+18{a}^{2} 5p420p315p25{p}^{4}-20{p}^{3}-15{p}^{2} 5p2(p2 − 4p − 3) 11x6+44x5121x411{x}^{6}+44{x}^{5}-121{x}^{4} 8c5+40c456c38{c}^{5}+40{c}^{4}-56{c}^{3} 8c3(c2 + 5c − 7) 3n24-3n - 24 7p84-7p - 84 −7(p + 12) 15a240a-15{a}^{2}-40a 18b266b-18{b}^{2}-66b −6b(3b + 11) 10y3+60y2-10{y}^{3}+60{y}^{2} 8a3+32a2-8{a}^{3}+32{a}^{2} −8a2(a − 4) 4u5+56u3-4{u}^{5}+56{u}^{3} 9b5+63b3-9{b}^{5}+63{b}^{3} −9b3(b2 − 7)

Everyday Math

Revenue A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of pp dollars each is given by the polynomial 5p2+150p-5{p}^{2}+150p. Factor the greatest common factor from this polynomial. Height of a baseball The height of a baseball hit with velocity 8080 feet/second at 44 feet above ground level is 16t2+80t+4-16{t}^{2}+80t+4, with t=t= the number of seconds since it was hit. Factor the greatest common factor from this polynomial. −4(4t2 − 20t − 1)

Writing Exercises

The greatest common factor of 3636 and 6060 is 1212. Explain what this means. What is the GCF of y4{y}^{4} , y5{y}^{5} , and y10{y}^{10} ? Write a general rule that tells how to find the GCF of ya{y}^{\text{a}} , yb{y}^{\text{b}} , and yc{y}^{\text{c}} .

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter Review Exercises

Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. y2+8y20{y}^{2}+8y - 20 trinomial 6a4-6{a}^{4} 9x319{x}^{3}-1 binomial n33n2+3n1{n}^{3}-3{n}^{2}+3n - 1 Determine the Degree of Polynomials In the following exercises, determine the degree of each polynomial. 16x240x2516{x}^{2}-40x - 25 2 5m+95m+9 15-15 0 y2+6y3+9y4{y}^{2}+6{y}^{3}+9{y}^{4} Add and Subtract Monomials In the following exercises, add or subtract the monomials. 4p+11p4p+11p 15p 8y35y3-8{y}^{3}-5{y}^{3} Add 4n5,-n5,6n54{n}^{5},\text{-}{n}^{5},-6{n}^{5} −3n5 Subtract 10x210{x}^{2} from 3x23{x}^{2} Add and Subtract Polynomials In the following exercises, add or subtract the polynomials. (4a2+9a11)+(6a25a+10)\left(4{a}^{2}+9a - 11\right)+\left(6{a}^{2}-5a+10\right) 10a2 + 4a − 1 (8m2+12m5)(2m27m1)\left(8{m}^{2}+12m - 5\right)-\left(2{m}^{2}-7m - 1\right) (y23y+12)+(5y29)\left({y}^{2}-3y+12\right)+\left(5{y}^{2}-9\right) 6y2 − 3y + 3 (5u2+8u)(4u7)\left(5{u}^{2}+8u\right)-\left(4u - 7\right) Find the sum of 8q3278{q}^{3}-27 and q2+6q2{q}^{2}+6q - 2 8q3 + q2 + 6q − 29 Find the difference of x2+6x+8{x}^{2}+6x+8 and x28x+15{x}^{2}-8x+15 Evaluate a Polynomial for a Given Value of the Variable In the following exercises, evaluate each polynomial for the given value. 200x15x2[/latex]when[latex]x=5200x-\frac{1}{5}{x}^{2}[/latex] when [latex]x=5 995 200x15x2[/latex]when[latex]x=0200x-\frac{1}{5}{x}^{2}[/latex] when [latex]x=0 200x15x2[/latex]when[latex]x=15200x-\frac{1}{5}{x}^{2}[/latex] when [latex]x=15 2,955 5+40x12x2[/latex]when[latex]x=105+40x-\frac{1}{2}{x}^{2}[/latex] when [latex]x=10 5+40x12x2[/latex]when[latex]x=45+40x-\frac{1}{2}{x}^{2}[/latex] when [latex]x=-4 −163 5+40x12x2[/latex]when[latex]x=05+40x-\frac{1}{2}{x}^{2}[/latex] when [latex]x=0 A pair of glasses is dropped off a bridge 640640 feet above a river. The polynomial 16t2+640-16{t}^{2}+640 gives the height of the glasses tt seconds after they were dropped. Find the height of the glasses when t=6t=6. 64 feet The fuel efficiency (in miles per gallon) of a bus going at a speed of xx miles per hour is given by the polynomial 1160x2+12x-\frac{1}{160}{x}^{2}+\frac{1}{2}x. Find the fuel efficiency when x=20x=20 mph.

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents In the following exercises, simplify. 63{6}^{3} 216 (12)4{\left(\frac{1}{2}\right)}^{4} (0.5)2{\left(-0.5\right)}^{2} 0.25 32-{3}^{2} Simplify Expressions Using the Product Property of Exponents In the following exercises, simplify each expression. p3p10{p}^{3}\cdot {p}^{10} p13 2262\cdot {2}^{6} aa2a3a\cdot {a}^{2}\cdot {a}^{3} a6 xx8x\cdot {x}^{8} Simplify Expressions Using the Power Property of Exponents In the following exercises, simplify each expression. (y4)3{\left({y}^{4}\right)}^{3} y12 (r3)2{\left({r}^{3}\right)}^{2} (32)5{\left({3}^{2}\right)}^{5} 310 (a10)y{\left({a}^{10}\right)}^{y} Simplify Expressions Using the Product to a Power Property In the following exercises, simplify each expression. (8n)2{\left(8n\right)}^{2} 64n2 (5x)3{\left(-5x\right)}^{3} (2ab)8{\left(2ab\right)}^{8} 256a8b8 (10mnp)4{\left(-10mnp\right)}^{4} Simplify Expressions by Applying Several Properties In the following exercises, simplify each expression. (3a5)3{\left(3{a}^{5}\right)}^{3} 27a15 (4y)2(8y){\left(4y\right)}^{2}\left(8y\right) (x3)5(x2)3{\left({x}^{3}\right)}^{5}{\left({x}^{2}\right)}^{3} x21 (5st2)3(2s3t4)2{\left(5s{t}^{2}\right)}^{3}{\left(2{s}^{3}{t}^{4}\right)}^{2} Multiply Monomials In the following exercises, multiply the monomials. (6p4)(9p)\left(-6{p}^{4}\right)\left(9p\right) −54p5 (13c2)(30c8)\left(\frac{1}{3}{c}^{2}\right)\left(30{c}^{8}\right) (8x2y5)(7xy6)\left(8{x}^{2}{y}^{5}\right)\left(7x{y}^{6}\right) 56x3y11 (23m3n6)(16m4n4)\left(\frac{2}{3}{m}^{3}{n}^{6}\right)\left(\frac{1}{6}{m}^{4}{n}^{4}\right)

Multiply Polynomials

Multiply a Polynomial by a Monomial In the following exercises, multiply. 7(10x)7\left(10-x\right) 70 − 7x a2(a29a36){a}^{2}\left({a}^{2}-9a - 36\right) 5y(125y31)-5y\left(125{y}^{3}-1\right) −625y4 + 5y (4n5)(2n3)\left(4n - 5\right)\left(2{n}^{3}\right) Multiply a Binomial by a Binomial In the following exercises, multiply the binomials using various methods. (a+5)(a+2)\left(a+5\right)\left(a+2\right) a2 + 7a + 10 (y4)(y+12)\left(y - 4\right)\left(y+12\right) (3x+1)(2x7)\left(3x+1\right)\left(2x - 7\right) 6x2 − 19x − 7 (6p11)(3p10)\left(6p - 11\right)\left(3p - 10\right) (n+8)(n+1)\left(n+8\right)\left(n+1\right) n2 + 9n + 8 (k+6)(k9)\left(k+6\right)\left(k - 9\right) (5u3)(u+8)\left(5u - 3\right)\left(u+8\right) 5u2 + 37u − 24 (2y9)(5y7)\left(2y - 9\right)\left(5y - 7\right) (p+4)(p+7)\left(p+4\right)\left(p+7\right) p2 + 11p + 28 (x8)(x+9)\left(x - 8\right)\left(x+9\right) (3c+1)(9c4)\left(3c+1\right)\left(9c - 4\right) 27c2 − 3c − 4 (10a1)(3a3)\left(10a - 1\right)\left(3a - 3\right) Multiply a Trinomial by a Binomial In the following exercises, multiply using any method. (x+1)(x23x21)\left(x+1\right)\left({x}^{2}-3x - 21\right) x3 − 2x2 − 24x − 21 (5b2)(3b2+b9)\left(5b - 2\right)\left(3{b}^{2}+b - 9\right) (m+6)(m27m30)\left(m+6\right)\left({m}^{2}-7m - 30\right) m3m2 − 72m − 180 (4y1)(6y212y+5)\left(4y - 1\right)\left(6{y}^{2}-12y+5\right)

Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents In the following exercises, simplify. 2822\frac{{2}^{8}}{{2}^{2}} 26or 64 a6a\frac{{a}^{6}}{a} n3n12\frac{{n}^{3}}{{n}^{12}} 1n9\frac{1}{{n}^{9}} xx5\frac{x}{{x}^{5}} Simplify Expressions with Zero Exponents In the following exercises, simplify. 30{3}^{0} 1 y0{y}^{0} (14t)0{\left(14t\right)}^{0} 1 12a015b012{a}^{0}-15{b}^{0} Simplify Expressions Using the Quotient to a Power Property In the following exercises, simplify. (35)2{\left(\frac{3}{5}\right)}^{2} 925\frac{9}{25} (x2)5{\left(\frac{x}{2}\right)}^{5} (5mn)3{\left(\frac{5m}{n}\right)}^{3} 125m3n3\frac{125{m}^{3}}{{n}^{3}} (s10t)2{\left(\frac{s}{10t}\right)}^{2} Simplify Expressions by Applying Several Properties In the following exercises, simplify. (a3)2a4\frac{{\left({a}^{3}\right)}^{2}}{{a}^{4}} a2 u3u2u4\frac{{u}^{3}}{{u}^{2}\cdot {u}^{4}} (xx9)5{\left(\frac{x}{{x}^{9}}\right)}^{5} 1x40\frac{1}{{x}^{40}} (p4p5p3)2{\left(\frac{{p}^{4}\cdot {p}^{5}}{{p}^{3}}\right)}^{2} (n5)3(n2)8\frac{{\left({n}^{5}\right)}^{3}}{{\left({n}^{2}\right)}^{8}} 1n\frac{1}{n} (5s24t)3{\left(\frac{5{s}^{2}}{4t}\right)}^{3} Divide Monomials In the following exercises, divide the monomials. 72p12÷8p372{p}^{12}\div 8{p}^{3} 9p9 26a8÷(2a2)-26{a}^{8}\div \left(2{a}^{2}\right) 45y615y10\frac{45{y}^{6}}{-15{y}^{10}} 3y4-\frac{3}{{y}^{4}} 30x836x9\frac{-30{x}^{8}}{-36{x}^{9}} 28a9b7a4b3\frac{28{a}^{9}b}{7{a}^{4}{b}^{3}} 4a5b2\frac{4{a}^{5}}{{b}^{2}} 11u6v355u2v8\frac{11{u}^{6}{v}^{3}}{55{u}^{2}{v}^{8}} (5m9n3)(8m3n2)(10mn4)(m2n5)\frac{\left(5{m}^{9}{n}^{3}\right)\left(8{m}^{3}{n}^{2}\right)}{\left(10m{n}^{4}\right)\left({m}^{2}{n}^{5}\right)} 4m9n4\frac{4{m}^{9}}{{n}^{4}} 42r2s46rs354rs29s\frac{42{r}^{2}{s}^{4}}{6r{s}^{3}}-\frac{54r{s}^{2}}{9s}

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent In the following exercises, simplify. 62{6}^{-2} 136\frac{1}{36} (10)3{\left(-10\right)}^{-3} 5245\cdot {2}^{-4} 516\frac{5}{16} (8n)1{\left(8n\right)}^{-1} Simplify Expressions with Integer Exponents In the following exercises, simplify. x3x9{x}^{-3}\cdot {x}^{9} x6 r5r4{r}^{-5}\cdot {r}^{-4} (uv3)(u4v2)\left(u{v}^{-3}\right)\left({u}^{-4}{v}^{-2}\right) 1u3v5\frac{1}{{u}^{3}{v}^{5}} (m5)1{\left({m}^{5}\right)}^{-1} (k2)3{\left({k}^{-2}\right)}^{-3} k6 q4q20\frac{{q}^{4}}{{q}^{20}} b8b2\frac{{b}^{8}}{{b}^{-2}} b10 n3n5\frac{{n}^{-3}}{{n}^{-5}} Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 5,300,0005,300,000 5.3 × 106 0.008140.00814 The thickness of a piece of paper is about 0.0970.097 millimeter. 9.7 × 10−2 According to www.cleanair.com, U.S. businesses use about 21,000,00021,000,000 tons of paper per year. Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 2.9×1042.9\times {10}^{4} 29,000 1.5×1081.5\times {10}^{8} 3.75×1013.75\times {10}^{-1} 375 9.413×1059.413\times {10}^{-5} Multiply and Divide Using Scientific Notation In the following exercises, multiply and write your answer in decimal form. (3×107)(2×104)\left(3\times {10}^{7}\right)\left(2\times {10}^{-4}\right) 6,000 (1.5×103)(4.8×101)\left(1.5\times {10}^{-3}\right)\left(4.8\times {10}^{-1}\right) 6×1092×101\frac{6\times {10}^{9}}{2\times {10}^{-1}} 30,000,000,000 9×1031×106\frac{9\times {10}^{-3}}{1\times {10}^{-6}}

Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions In the following exercises, find the greatest common factor. 5n,455n,45 5 8a,728a,72 12x2,20x3,36x412{x}^{2},20{x}^{3},36{x}^{4} 4x2 9y4,21y5,15y69{y}^{4},21{y}^{5},15{y}^{6} Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. 16u2416u - 24 8(2u − 3) 15r+3515r+35 6p2+6p6{p}^{2}+6p 6p(p + 1) 10c210c10{c}^{2}-10c 9a59a3-9{a}^{5}-9{a}^{3} −9a3(a2 + 1) 7x828x3-7{x}^{8}-28{x}^{3} 5y255y+455{y}^{2}-55y+45 5(y2 − 11y + 9) 2q516q3+30q22{q}^{5}-16{q}^{3}+30{q}^{2}

Chapter Practice Test

For the polynomial 8y43y2+18{y}^{4}-3{y}^{2}+1
  1. ⓐ Is it a monomial, binomial, or trinomial?
  2. ⓑ What is its degree?
  1. ⓐ trinomial
  2. ⓑ 4
In the following exercises, simplify each expression. (5a2+2a12)+(9a2+8a4)\left(5{a}^{2}+2a - 12\right)+\left(9{a}^{2}+8a - 4\right) (10x23x+5)(4x26)\left(10{x}^{2}-3x+5\right)-\left(4{x}^{2}-6\right) 6x2 − 3x + 11 (34)3{\left(-\frac{3}{4}\right)}^{3} nn4n\cdot {n}^{4} n5 (10p3q5)2{\left(10{p}^{3}{q}^{5}\right)}^{2} (8xy3)(6x4y6)\left(8x{y}^{3}\right)\left(-6{x}^{4}{y}^{6}\right) −48x5y9 4u(u29u+1)4u\left({u}^{2}-9u+1\right) (s+8)(s+9)\left(s+8\right)\left(s+9\right) s2 + 17s + 72 (m+3)(7m2)\left(m+3\right)\left(7m - 2\right) (11a6)(5a1)\left(11a - 6\right)\left(5a - 1\right) 55a2 − 41a + 6 (n8)(n24n+11)\left(n - 8\right)\left({n}^{2}-4n+11\right) (4a+9b)(6a5b)\left(4a+9b\right)\left(6a - 5b\right) 24a2 + 34ab − 45b2 5658\frac{{5}^{6}}{{5}^{8}} (x3x9x5)2{\left(\frac{{x}^{3}\cdot {x}^{9}}{{x}^{5}}\right)}^{2} x14 (47a18b23c5)0{\left(47{a}^{18}{b}^{23}{c}^{5}\right)}^{0} 24r3s6r2s7\frac{24{r}^{3}s}{6{r}^{2}{s}^{7}} 4rs6\frac{4r}{{s}^{6}} 8y216y+204y\frac{8{y}^{2}-16y+20}{4y} (15xy335x2y)÷5xy\left(15x{y}^{3}-35{x}^{2}y\right)\div 5xy 3y2 − 7x 41{4}^{-1} (2y)3{\left(2y\right)}^{-3} 12y\frac{1}{2y} p3p8{p}^{-3}\cdot {p}^{-8} x4x5\frac{{x}^{4}}{{x}^{-5}} x9 (2.4×108)(2×105)\left(2.4\times {10}^{8}\right)\left(2\times {10}^{-5}\right) In the following exercises, factor the greatest common factor from each polynomial. 80a3+120a2+40a80{a}^{3}+120{a}^{2}+40a 6x230x-6{x}^{2}-30x −6x(x + 5) Convert 5.25×1045.25\times {10}^{-4} to decimal form. 0.000525 In the following exercises, simplify, and write your answer in decimal form. 9×1043×101\frac{9\times {10}^{4}}{3\times {10}^{-1}} 3 × 105 A hiker drops a pebble from a bridge 240240 feet above a canyon. The polynomial 16t2+240-16{t}^{2}+240 gives the height of the pebble tt seconds a after it was dropped. Find the height when t=3t=3. According to www.cleanair.org, the amount of trash generated in the US in one year averages out to 112,000112,000 pounds of trash per person. Write this number in scientific notation.

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