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# Key Concepts & Glossary

## 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, 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 $f\left(k\right)\\$