Key Concepts & Glossary
Key Concepts
- To find , determine the remainder of the polynomial when it is divided by .
- k is a zero of if and only if is a factor of .
- Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient.
- When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
- Synthetic division can be used to find the zeros of a polynomial function.
- According to the Fundamental Theorem, every polynomial function has at least one complex zero.
- Every polynomial function with degree greater than 0 has at least one complex zero.
- Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form , where c is a complex number.
- The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
- The number of negative real zeros of a polynomial function is either the number of sign changes of or less than the number of sign changes by an even integer.
- Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.
Glossary
- Descartes’ Rule of Signs
- a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of and
- Factor Theorem
- k is a zero of polynomial function if and only if is a factor of
- Fundamental Theorem of Algebra
- a polynomial function with degree greater than 0 has at least one complex zero
- Linear Factorization Theorem
- allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form , where c is a complex number
- Rational Zero Theorem
- the possible rational zeros of a polynomial function have the form where p is a factor of the constant term and q is a factor of the leading coefficient.
- Remainder Theorem
- if a polynomial is divided by , then the remainder is equal to the value