Key Concepts

• To find $f\left(k\right)$, determine the remainder of the polynomial $f\left(x\right)$ when it is divided by $x-k$.
• k is a zero of $f\left(x\right)$ if and only if $\left(x-k\right)$  is a factor of $f\left(x\right)$.
• 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 of Algebra, 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 $\left(x-c\right)$ 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 $f\left(-x\right)$  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 $f\left(x\right)$ and $f\left(-x\right)$

Factor Theorem

k is a zero of polynomial function $f\left(x\right)$ if and only if $\left(x-k\right)$  is a factor of $f\left(x\right)$

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 $\left(x-c\right)$ where c is a complex number

Rational Zero Theorem

the possible rational zeros of a polynomial function have the form $\frac{p}{q}$ where p is a factor of the constant term and q is a factor of the leading coefficient

Remainder Theorem

if a polynomial $f\left(x\right)$ is divided by $x-k$ , then the remainder is equal to the value $f\left(k\right)$