# Polynomial and Synthetic Division

## Section 3.3 Objectives

• Use long division to divide polynomials by other

polynomials.

• Use synthetic division to divide polynomials by

binomials of the form (x – k).

• Use the Remainder Theorem and the Factor

Theorem.

## Long Division

Find the quotient and remainder when

2x^{3} − 8x^{2} + x − 4 is divided by 2x^{2} −1

Check:

Thus,

## Division Algorithm for Polynomials

If f(x) and d(x) denote polynomial functions and if d(x)
is

not the zero polynomial, then there are unique

polynomials q(x) and r(x) such that

where r(x) is either the zero polynomial or a

polynomial of degree less than that of d(x). If the

remainder r(x) is zero, d(x) divides evenly into f(x).

## Synthetic Division

Use synthetic division to find the quotient and

remainder when

Coefficients of the dividend in descending powers of x are:

3 0 -1 11

Use the usual division sign and enter -2 in front of

it (since the divisor is x + 2)

Multiply the latest entry in the bottom row by -2

and place the result in row 2 one column to the

right.

Add the last entry to the above number and enter

the sum into the bottom row.

Repeat

## Remainder Theorem

Let f be a polynomial function. If f(x) is

divided by x - c, then the remainder is f(c).

Find the remainder if

is divided by x + 3.

x + 3 = x - (-3)

## Factor Theorem

1. If f(c)=0, then x - c is a factor of f(x).

2. If x - c is a factor of f(x), then f(c)=0.

Use the Factor Theorem to determine whether the

function has the factor

x +3 is not a factor of f(x).

x + 4 is a factor of f(x).

## What you should have learned

• Use long division to divide polynomials by other

polynomials.

• Use synthetic division to divide polynomials by

binomials of the form (x – k).

• Use the Remainder Theorem and the Factor

Theorem.