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Objective Use long division and synthetic division to divide polynomials.

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

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**Example 1: Using Long Division to Divide a Polynomial**

Divide using long division. (–y2 + 2y3 + 25) ÷ (y – 3) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. 2y3 – y2 + 0y + 25 Step 2 Write division in the same way you would when dividing numbers. y – 3 2y3 – y2 + 0y + 25

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Example 1 Continued Step 3 Divide. 2y2 + 5y + 15 y – 3 2y3 – y2 + 0y + 25 –(2y3 – 6y2) 5y2 + 0y –(5y2 – 15y) 15y + 25 –(15y – 45) 70

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Example 1 Continued Step 4 Write the final answer. –y2 + 2y3 + 25 y – 3 = 2y2 + 5y 70 y – 3

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**Divide using long division.**

(15x2 + 8x – 12) ÷ (3x + 1)

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**Divide using long division.**

(x2 + 5x – 28) ÷ (x – 3)

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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).

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

Divide using synthetic division. (3x4 – x3 + 5x – 1) ÷ (x + 2) Step 1 Find a. a = –2 For (x + 2), a = –2. Step 2 Write the coefficients and a in the synthetic division format. 3 – –1 –2 Use 0 for the coefficient of x2.

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Example 2B Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. –2 3 – –1 Draw a box around the remainder, 45. –6 14 –28 46 3 –7 14 –23 45 Step 4 Write the quotient. 3x3 – 7x2 + 14x – 23 + 45 x + 2 Write the remainder over the divisor.

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**Divide using synthetic division.**

(6x2 – 5x – 6) ÷ (x + 3) Step 1 Find a. a = –3 For (x + 3), a = –3. Step 2 Write the coefficients and a in the synthetic division format. –3 6 –5 –6 Write the coefficients of 6x2 – 5x – 6.

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**Step 3 Bring down the first coefficient**

Step 3 Bring down the first coefficient. Then multiply and add for each column. –3 6 –5 –6 Draw a box around the remainder, 63. –18 69 6 –23 63 Step 4 Write the quotient. 6x – 23 + 63 x + 3 Write the remainder over the divisor.

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**Divide using synthetic division.**

(x2 – 3x – 18) ÷ (x – 6)

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**You can use synthetic division to evaluate polynomials**

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.

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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. 2 –1 7 Write the coefficients of the dividend. Use a = 2. 4 18 34 2 9 17 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

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

P(x) = x3 + 3x2 + 4 for x = –3.

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

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

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Lesson Quiz 1. Divide by using long division. (8x3 + 6x2 + 7) ÷ (x + 2) 8x2 – 10x + 20 – 33 x + 2 2. Divide by using synthetic division. (x3 – 3x + 5) ÷ (x + 2) x2 – 2x + 1 + 3 x + 2 3. Use synthetic substitution to evaluate P(x) = x3 + 3x2 – 6 for x = 5 and x = –1. 194; –4

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