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Holt McDougal Algebra 2 6-3 Dividing Polynomials Warm Up Divide using long division. 1. 12.18 ÷ 2.1 2. 3. 2x + 5y 5.8 7a b Divide. 6x – 15y 3 7a 2 ab a

Warm Up Divide using long division ÷ Divide

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Objective Use long division and synthetic division to divide polynomials.

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Page 1: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing Polynomials

Warm UpDivide using long division.1. 12.18 ÷ 2.1

2.

3.

2x + 5y

5.8

7a – b

Divide.

6x – 15y3

7a2 – aba

Page 2: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing Polynomials

Use long division and synthetic division to divide polynomials.

Objective

Page 3: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsPolynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

Page 4: Warm Up Divide using long division ÷ Divide

Holt McDougal 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, includingterms with a coefficient of 0.

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

y – 3 2y3 – y2 + 0y + 25

Page 5: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck It Out! Example 1a

Divide using long division. (15x2 + 8x – 12) ÷ (3x + 1)

15x2 + 8x – 12

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

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

3x + 1 15x2 + 8x – 12

Page 6: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck It Out! Example 1b

Divide using long division. (x2 + 5x – 28) ÷ (x – 3)

x2 + 5x – 28

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

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

x – 3 x2 + 5x – 28

Page 7: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing Polynomials

Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form (x – a).

Page 8: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing Polynomials

Page 9: Warm Up Divide using long division ÷ Divide

Holt McDougal 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.

13

For (x – ), a = .13

13

13a =

13

3 9 –2

Page 10: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsExample 2A Continued

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

Draw a box around the remainder, 1 .13

13

3 9 –2 1

3

Step 3 Write the quotient.

3x + 10 +1 1

313x –

10 131

133

Page 11: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsExample 2A Continued

3x + 10 +1 1

313x –

Check Multiply (x – ) 13

= 3x2 + 9x – 2

(x – ) 13(x – ) 1

3 (x – ) 133x + 10 +

1 1313x –

Page 12: Warm Up Divide using long division ÷ Divide

Holt McDougal 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.

Page 13: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsExample 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.

–63 45

Step 4 Write the quotient.

3x3 – 7x2 + 14x – 23 + 45x + 2

Write the remainder over the divisor.

46–2814–2314–7

Page 14: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck 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.

Page 15: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck 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 + 63x + 3

Write the remainder over the divisor.

–2369

Page 16: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck 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.

Page 17: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck 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

183

Page 18: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing PolynomialsCheck It Out! Example 3

Write an expression for the length of a rectangle with width y – 9 and area y2 – 14y + 45.

Substitute.

Use synthetic division.

The area A is related to the width w and the length l by the equation A = l w.

y2 – 14y + 45 y – 9l(x) =

1 –14 45 9 9

1 0The length of the rectangle can be represented by l(x)= y – 5.

–45–5

Page 19: Warm Up Divide using long division ÷ Divide

Holt McDougal Algebra 2

6-3 Dividing Polynomials

3. Find an expression for the height of a parallelogram whose area is represented by 2x3 – x2 – 20x + 3 and whose base is represented by (x + 3).

Lesson Quiz

2. Divide by using synthetic division. (x3 – 3x + 5) ÷ (x + 2)

1. Divide by using long division. (8x3 + 6x2 + 7) ÷ (x + 2)

2x2 – 7x + 1

8x2 – 10x + 20 – 33 x + 2

x2 – 2x + 1 + 3 x + 2