Warm UpDivide using long division.

1. 161 ÷ 7

2. 12.18 ÷ 2.1

3.

4.

2x + 5y

23

5.8

7a – b

Divide.

6x – 15y3

7a2 – aba

Use long division and synthetic division to divide polynomials.

Objective

synthetic division

Vocabulary

Polynomial long division is a method for dividing a polynomial by another polynomials of a lower degree. It is very similar to dividing numbers.

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).

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

13

a =

13

3 9 –2

Example 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

313

x –

10 131

133

Example 2A Continued

3x + 10 +1 1

313

x – Check Multiply (x – ) 1

3

= 3x2 + 9x – 2

(x – ) 13

(x – ) 13

(x – ) 133x + 10 +

1 1313

x –

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.

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 + 2Write the remainder over the divisor.

46–2814

–2314–7

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.

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 + 3Write the remainder over the divisor.

–23

69

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.

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

You can use synthetic division to evaluate polynomials. This process is called synthetic substitution. The process of synthetic substitution is exactly the same as the process of synthetic division, but the final answer is interpreted differently, as described by the Remainder Theorem.

Example 3A: Using Synthetic Substitution

Use synthetic substitution to evaluate the polynomial for the given value.

P(x) = 2x3 + 5x2 – x + 7 for x = 2.

Write the coefficients of the dividend. Use a = 2.

2 5 –1 7 2

4

2 41

P(2) = 41

Check Substitute 2 for x in P(x) = 2x3 + 5x2 – x + 7.P(2) = 2(2)3 + 5(2)2 – (2) + 7

P(2) = 41

3418

179

Example 3B: Using Synthetic Substitution

Use synthetic substitution to evaluate the polynomial for the given value.

P(x) = 6x4 – 25x3 – 3x + 5 for x = – .

6 –25 0 –3 5

–2

6 7

13

–13

Write the coefficients of the dividend. Use 0 for the coefficient of x2. Use a = . 1

3P( ) = 71

3

2–39

–69–27

Check It Out! Example 3a

Use synthetic substitution to evaluate the polynomial for the given value.

P(x) = x3 + 3x2 + 4 for x = –3.

Write the coefficients of the dividend. Use 0 for the coefficient of x2 Use a = –3.

1 3 0 4 –3

–3

1 4

P(–3) = 4

Check Substitute –3 for x in P(x) = x3 + 3x2 + 4.P(–3) = (–3)3 + 3(–3)2 + 4

P(–3) = 4

00

00

Check It Out! Example 3b

Use synthetic substitution to evaluate the polynomial for the given value.

P(x) = 5x2 + 9x + 3 for x = .

5 9 3

1

5 5

151

5 Write the coefficients of the dividend. Use a = . 1

5

P( ) = 515

2

10

Example 4: Geometry ApplicationWrite an expression that represents the area of the top face of a rectangular prism when the height is x + 2 and the volume of the prism is x3 – x2 – 6x.

Substitute.

Use synthetic division.

The volume V is related to the area A and the height h by the equation V = A h. Rearrangingfor A gives A = . V

hx3 – x2 – 6x

x + 2A(x) =

1 –1 –6 0 –2

–2

1 0

The area of the face of the rectangular prism can be represented by A(x)= x2 – 3x.

06

0–3

Check It Out! Example 4Write 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 – 9

l(x) =

1 –14 45 9

9

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

–45

–5

4. 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

194; –43. Use synthetic substitution to evaluate P(x) = x3 + 3x2 – 6 for x = 5 and x = –1.

8x2 – 10x + 20 – 33

x + 2

x2 – 2x + 1 + 3

x + 2

Homework!

Holt Chapter 6 Sec 3Page 426 #19, 23, 25, 27, 29,

33, 38, 45, 49, 50, 57, 59