3­3 Dividing Polynomials­ period 1.notebook

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Bellwork 10-17-14

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

3­3 Dividing Polynomials­ period 1.notebook

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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).

3­3 Dividing Polynomials­ period 1.notebook

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3­3 Dividing Polynomials­ period 1.notebook

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Divide using synthetic division.

Example 2A: Using Synthetic Division to Divide by a Linear Binomial

(3x2 + 9x – 2) ÷ (x – ) 13

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

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

For (x – ), a = .13

13

13a =

13

3 9 –2

3­3 Dividing Polynomials­ period 1.notebook

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Example 2A ContinuedStep 2 Bring down the first coefficient. Then multiply and add for each column.

Draw a box around the remainder.

13

3 9 –2

Step 3 Write the quotient.

3­3 Dividing Polynomials­ period 1.notebook

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Example 2A Continued

3x + 10 +1

31x –

1

3Check Multiply (x – ) 1

3

(x – ) 13(x – ) 1

3 (x – ) 133x + 10 +

1313x –

1

= 3x2 + 9x – 2   

3­3 Dividing Polynomials­ period 1.notebook

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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

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

3­3 Dividing Polynomials­ period 1.notebook

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Example 2B ContinuedStep 3 Bring down the first coefficient. Then multiply and add for each column.

Step 4 Write the quotient.

3 – 1 0 5 –1 –2

3­3 Dividing Polynomials­ period 1.notebook

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Check It Out! Example 2aDivide using synthetic division.

(6x2 – 5x – 6) ÷ (x + 3)

Step 1 Find a.

Write the coefficients of 6x2 – 5x – 6.–3 6 –5 –6

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

For (x + 3), a = –3

3­3 Dividing Polynomials­ period 1.notebook

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Check It Out! Example 2a ContinuedStep 3 Bring down the first coefficient. Then multiply and add for each column.

Step 4 Write the quotient.

–3 6 –5 –6

3­3 Dividing Polynomials­ period 1.notebook

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Check It Out! Example 2bDivide using synthetic division.

(x2 – 3x – 18) ÷ (x – 6)

Step 1 Find a.

Write the coefficients of x2 – 3x – 18.

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

3­3 Dividing Polynomials­ period 1.notebook

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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.

3­3 Dividing Polynomials­ period 1.notebook

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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

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

3­3 Dividing Polynomials­ period 1.notebook

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Example 3B: Using Synthetic SubstitutionUse synthetic substitution to evaluate the polynomial for the given value.P(x) = 6x4 – 25x3 – 3x + 5 for x = – .1

3Write the coefficients of the dividend. Use 0 for the coefficient of x2. Use a = . 1

3

3­3 Dividing Polynomials­ period 1.notebook

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Check It Out! Example 3aUse 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.

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

P(–3) = (–3)3 + 3(–3)2 + 4

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Check It Out! Example 3bUse synthetic substitution to evaluate the polynomial for the given value.

P(x) = 5x2 + 9x + 3 for x = .15

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

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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 Quiz1. Divide by using synthetic division. (x3 – 3x + 5) ÷ (x + 2)

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

2x2 – 7x + 1

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Homework: Workbook Page 133, #'s 1-11 all