3.7 Warm UpSolve the equation.

1. √(x + 3) + 8 = 15

2. √(8 – 3x) + 5 = 6

3. 4√(2x + 1) – 7 = 1

3.7 Divide Polynomials

To divide a polynomial by a monomial

Write each term as a fraction

Divide each term separately

Simplify

SOLUTION

Divide a polynomial by a monomial

EXAMPLE 1

Divide 4x3 + 8x2 + 10x by 2x.

Method 1: Write the division as a fraction.

Write as fraction.

Divide each term by 2x.

= 2x2 + 4x + 5

8x24x3

2x 2x10x2x+ +=

Simplify.

(4x3 + 8x2 + 10x) 2x 4x3 + 8x2 +10x2x=

GUIDED PRACTICE for Example 1

(6x3 + 3x2 –12x) 3x1.

2x2 + x – 4

ANSWER

(12y4 – 16y3 + 20y2) 4y2.

ANSWER

3y3 – 4y2 + 5y

Divide polynomial by a binomial Divide the 1st term of the polynomial

by the 1st term of the binomial (step 1) Multiply the whole binomial by step 1 Subtract from polynomial as in long

division Continue this pattern Remainder will be placed over the

binomial

SOLUTION

EXAMPLE 2 Divide a polynomial by a binomial

Divide x2 + 2x – 3 by x – 1.

STEP 1Divide the first term of x2 + 2x – 3 by the first term of x – 1.

Multiply x and x – 1.

xx – 1 x2 + 2x – 3

x2 – x3x Subtract x2 – x from x2 + 2x.

Think: x2 x = ?

EXAMPLE 3 Divide a polynomial by a binomial

Divide 2x2 + 11x – 9 by 2x – 3.x

2x2 –3x2x – 3 2x2 + 11x – 9

14x – 9

12

Multiply x and 2x – 3.

Subtract 2x2 – 3x. Bring down – 9.

Multiply 7 and 2x – 3.

Subtract 14x – 21.

ANSWER (2x2 + 11x – 9) (2x – 3) = x + 7 + 12

2x – 3

14x – 21

+ 7

GUIDED PRACTICE for Examples 2 and 3

3. Divide: (a2 + 3a – 4) (a + 1)

a + 2 +ANSWER– 6a + 1

4. Divide: (9b2 + 6b + 8) (3b – 4)

ANSWER 3b + 6 +32

3b – 4

EXAMPLE 4 Rewrite polynomials

Divide 5y + y2 + 4 by 2 + y.

Rewrite polynomials.

Multiply y and y + 2.y2 + 2ySubtract y2 + 2y. Bring down 4.3y + 4Multiply 3 and y + 2.3y + 6Subtract 3y + 6.– 2

y + 2 y2 + 5y + 4y

ANSWER

(5y + y2 + 4) (2 + y) = y + 3 + y + 2– 2

+ 3

EXAMPLE 5 Insert missing terms

Divide 13 + 4m2 by – 1 + 2m.

Rewrite polynomials, Insert missing term.2m

2m – 1 4m2 + 0m + 13Multiply 2m and 2m – 1.4m2 – 2mSubtract 4m2 – 2m. Bring down 13.2m + 13

Multiply 1 and 2m – 1.2m – 1

Subtract 2m – 1.14

ANSWER

(13 + 4m2) (– 1 + 2m) = 2m + 1 + 2m – 114

+ 1

GUIDED PRACTICE for Examples 4, 5, and 6

5. Divide: (8m – 7 + 4m2) (5 + 2m)

ANSWER

2m – 1+2m + 5– 2

6. Divide: (n2 – 6) (– 3 + n)

n + 3 +

ANSWER

n – 3 3

EXAMPLE 6 Rewrite and graph a rational function

Graph y = 2x – 1x – 2

SOLUTION

STEP 1

+x – ha

Rewrite the rational function inthe form y = k.

x – 2 2x – 12x – 4

3

2

So, y = + 2.x – 23

EXAMPLE 6 Rewrite and graph a rational function

STEP 2

Graph the function.

Graph y = 3x + 1x + 1

GUIDED PRACTICE for Examples 4, 5, and 6