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# Real Zeros of Polynomials

## Remainder, Factor, and Rational Zero Theorems and Descartes' Rule of Signs.

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Real Zeros of Polynomials

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### Vocabulary Language: English

Descartes' Rule of Signs

Descartes' rule of signs is a technique for determining the number of positive and negative real roots of a polynomial.

factor theorem

The factor theorem states that if $f(x)$ is a polynomial of degree $n>0$ and $f(c)=0$, then $x-c$ is a factor of the polynomial $f(x)$.

factorization theorem

The factorization theorem states that If $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$, where $a_{n} \ne 0$, and $n$ is a positive integer, then $f(x)=a_{n}(x-c_{1})(x-c_{2})\cdots(x-c_{0})$ where the numbers $c_{i}$ are complex numbers.

Multiplicity

The multiplicity of a term describes the number of times the given term acts as a zero of the given function.

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

Rational Zero Theorem

The rational zero theorem states that for a polynomial, $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_n, a_{n-1}, \cdots a_0$ are integers, the rational roots can be determined from the factors of $a_n$ and $a_0$. More specifically, if $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$, then all the rational factors will have the form $\pm \frac{p}{q}$.

Remainder Theorem

The remainder theorem states that if $f(k) = r$, then $r$ is the remainder when dividing $f(x)$ by $(x - k)$.

Roots

The roots of a function are the values of x that make y equal to zero.

Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.

Zeroes

The zeroes of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.

Zeros

The zeros of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.