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8: Dividing Polynomials

Last updated

May 2, 2022
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- 7.3: Exercises
- 8.1: Long division

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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We now start our discussion of specific classes of examples of functions. The first class of functions that we discuss are polynomials and rational functions. Let us first recall the definition of polynomials and rational functions.

Definition: Monomial and Polynomial

A monomial is a number, a variable, or a product of numbers and variables. A polynomial is a sum (or difference) of monomials.

Example $\PageIndex{1}$

Examples of monomials and polynomials.

Solution

The following are examples of monomials:

$5,\quad x,\quad 7x^2 y,\quad -12 x^3y^2 z^4, \quad\sqrt{2}\cdot a^3 n^2 x y \nonumber$

The following are examples of polynomials:

$x^2+3x-7,\quad 4x^2y^3+2x+z^3+4mn^2,\quad -5x^3-x^2-4x-9, \quad 5x^2y^4 \nonumber$

In particular, every monomial is also a polynomial.

We are mainly interested in polynomials in one variable $x$ , and consider these as functions. For example, $f(x)=x^2+3x-7$ is such a function.

Definition: Polynomial Function

A polynomial is a function $f$ of the form

$f(x)=a_n x^n+a_{n-1} x^{n-1}+\dots +a_2 x^2+ a_1 x + a_0 \nonumber$

for some real (or complex) numbers $a_0, a_1,\dots, a_n$ . The domain of a polynomial $f$ is all real numbers (see our standard convention definition).

The numbers $a_0, a_1,\dots, a_n$ are called coefficients. For each $k$ , the number $a_k$ is the coefficient of $x^k$ . The number $a_n$ is called the leading coefficient and $n$ is the degree of the polynomial.

The zeros of a polynomial are usually referred to as roots. Therefore $x$ is a root of a polynomial $f$ precisely when $f(x)=0$ .

Definition: Rational Function

A rational function is a fraction of two polynomials $f(x)=\dfrac{g(x)}{h(x)}$ , where $g(x)$ and $h(x)$ are both polynomials. The domain of $f$ is all real numbers for which the denominator $h(x)$ is not zero:

$D_f\,\,=\,\,\{\,\, x \,\,| \,\, h(x)\neq 0 \,\, \} \nonumber$

The following are examples of rational functions:

$f(x)=\dfrac{-3x^2+7x-5}{2x^3+4x^2+3x+1}, \quad f(x)=\dfrac{1}{x}, \quad f(x)=-x^2+3x+5 \nonumber$

8.1: Long division
We now show how to divide two polynomials. The method is similar to the long division of natural numbers.
8.2: Dividing by (x - c)
8.3: Optional section- Synthetic division
8.4: Exercises

Definition: Monomial and Polynomial

Example 8.1\PageIndex{1}

Definition: Polynomial Function

Definition: Rational Function

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Example $\PageIndex{1}$