Holt Algebra 2

6-3 Dividing Polynomials

Polynomials can be divided using long division just

like you learned with numbers.

Divide) 214 ÷ 6

6 214 35

-18

34 -30

4 Remainder

214 ÷ 6 = 354

6

Holt Algebra 2

6-3 Dividing Polynomials

Holt Algebra 2

6-3 Dividing Polynomials

Divide using long division.

Example 1: Using Long Division to Divide a

Polynomial

(–y2 + 2y3 + 25) ÷ (y – 3)

2y3 – y2 + 0y + 25

Step 1 Write the dividend in standard form, including terms with a coefficient of 0.

Step 2 Write division in the same way you would when dividing numbers.

y – 3 2y3 – y2 + 0y + 25

Holt Algebra 2

6-3 Dividing Polynomials

Notice that y times 2y2 is 2y3.

Write 2y2 above 2y3.

Step 3 Divide.

2y2

–(2y3 – 6y2) Multiply y – 3 by 2y2. Then

subtract. Bring down the next

term. Divide 5y2 by y. 5y2 + 0y

+ 5y

–(5y2 – 15y) Multiply y – 3 by 5y. Then

subtract. Bring down the next

term. Divide 15y by y. 15y + 25

–(15y – 45)

70 Find the remainder.

+ 15

Multiply y – 3 by 15. Then

subtract.

Example 1 Continued

y – 3 2y3 – y2 + 0y + 25

Holt Algebra 2

6-3 Dividing Polynomials

Step 4 Write the final answer.

Example 1 Continued

–y2 + 2y3 + 25

y – 3 = 2y2 + 5y + 15 + 70

y – 3

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 1a

Divide using long division.

(15x2 + 8x – 12) ÷ (3x + 1)

15x2 + 8x – 12

Step 1 Write the dividend in standard form, including terms with a coefficient of 0.

Step 2 Write division in the same way you would when dividing numbers.

3x + 1 15x2 + 8x – 12

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 1a Continued

Notice that 3x times 5x is 15x2.

Write 5x above 15x2.

Step 3 Divide.

5x

–(15x2 + 5x) Multiply 3x + 1 by 5x. Then

subtract. Bring down the next

term. Divide 3x by 3x. 3x – 12

+ 1

–(3x + 1)

–13 Find the remainder.

Multiply 3x + 1 by 1. Then

subtract.

3x + 1 15x2 + 8x – 12

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 1a Continued

Step 4 Write the final answer.

15x2 + 8x – 12

3x + 1 = 5x + 1 – 13

3x + 1

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 1b

Divide using long division.

(x2 + 5x – 28) ÷ (x – 3)

x2 + 5x – 28

Step 1 Write the dividend in standard form, including terms with a coefficient of 0.

Step 2 Write division in the same way you would when dividing numbers.

x – 3 x2 + 5x – 28

Holt Algebra 2

6-3 Dividing Polynomials

Notice that x times x is x2.

Write x above x2.

Step 3 Divide.

x

–(x2 – 3x) Multiply x – 3 by x. Then

subtract. Bring down the next

term. Divide 8x by x. 8x – 28

+ 8

–(8x – 24)

–4 Find the remainder.

Multiply x – 3 by 8. Then

subtract.

Check It Out! Example 1b Continued

x – 3 x2 + 5x – 28

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 1b Continued

Step 4 Write the final answer.

x2 + 5x – 28

x – 3 = x + 8 –

4

x – 3

Holt Algebra 2

6-3 Dividing Polynomials

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial (x – a). The divisor must be a binomial. The x must have a coefficient of 1. Use the opposite sign of “a.”

Holt Algebra 2

6-3 Dividing Polynomials

Holt Algebra 2

6-3 Dividing Polynomials

Divide using synthetic division.

Example 2A: Using Synthetic Division to Divide by a

Linear Binomial

(3x2 + 9x – 2) ÷ (x – )

Step 1 Find a. Then write the coefficients and a in the synthetic division format.

Write the coefficients of 3x2 + 9x – 2.

1 3

For (x – ), a = . 1 3

1 3

1 3

a =

1 3

3 9 –2

Holt Algebra 2

6-3 Dividing Polynomials

Example 2A Continued

Step 2 Bring down the first coefficient. Then multiply and add for each column.

Draw a box around the remainder, 1 . 1 3

1 3

3 9 –2

1

3

Step 3 Write the quotient.

3x + 10 + 1

1 3 1 3

x –

10 1 3

1

1 3

3

Holt Algebra 2

6-3 Dividing Polynomials

Example 2A Continued

3x + 10 + 1

1 3 1 3

x – Check Multiply (x – )

1 3

= 3x2 + 9x – 2

(x – ) 1 3

(x – ) 1 3

(x – ) 1 3

3x + 10 +

1 1 3 1 3

x –

Holt Algebra 2

6-3 Dividing Polynomials

Divide using synthetic division.

(3x4 – x3 + 5x – 1) ÷ (x + 2)

Step 1 Find a.

Use 0 for the coefficient

of x2.

For (x + 2), a = –2. a = –2

Example 2B: Using Synthetic Division to Divide by a

Linear Binomial

3 – 1 0 5 –1 –2

Step 2 Write the coefficients and a in the synthetic division format.

Holt Algebra 2

6-3 Dividing Polynomials

Example 2B Continued

Draw a box around the

remainder, 45.

3 –1 0 5 –1 –2

Step 3 Bring down the first coefficient. Then multiply and add for each column.

–6

3 45

Step 4 Write the quotient.

3x3 – 7x2 + 14x – 23 + 45

x + 2 Write the remainder over

the divisor.

46 –28 14

–23 14 –7

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 2a

Divide using synthetic division.

(6x2 – 5x – 6) ÷ (x + 3)

Step 1 Find a.

Write the coefficients of 6x2 – 5x – 6.

For (x + 3), a = –3. a = –3

–3 6 –5 –6

Step 2 Write the coefficients and a in the synthetic division format.

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 2a Continued

Draw a box around the

remainder, 63.

6 –5 –6 –3

Step 3 Bring down the first coefficient. Then multiply and add for each column.

–18

6 63

Step 4 Write the quotient.

6x – 23 + 63

x + 3 Write the remainder over

the divisor.

–23

69

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 2b

Divide using synthetic division.

(x2 – 3x – 18) ÷ (x – 6)

Step 1 Find a.

Write the coefficients of x2 – 3x – 18.

For (x – 6), a = 6. a = 6

6 1 –3 –18

Step 2 Write the coefficients and a in the synthetic division format.

Holt Algebra 2

6-3 Dividing Polynomials

Check It Out! Example 2b Continued

There is no remainder. 1 –3 –18 6

Step 3 Bring down the first coefficient. Then multiply and add for each column.

6

1 0

Step 4 Write the quotient.

x + 3

18

3

Holt Algebra 2

6-3 Dividing Polynomials

Homework pg. 426

#’s 19-29 odd, 37-43 odd