Warm Up Divide using long division. 1. 12.18 ÷ 2.1 5.8 Divide. 6x – 15y 3 2. 2x + 5y 7a2 – ab a 3. 7a – b

Objective Use long division and synthetic division to divide polynomials.

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

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

Check It Out! Example 1a Divide using long division. (15x2 + 8x – 12) ÷ (3x + 1) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. 15x2 + 8x – 12 Step 2 Write division in the same way you would when dividing numbers. 3x + 1 15x2 + 8x – 12

Check It Out! Example 1b Divide using long division. (x2 + 5x – 28) ÷ (x – 3) Step 1 Write the dividend in standard form, including terms with a coefficient of 0. x2 + 5x – 28 Step 2 Write division in the same way you would when dividing numbers. x – 3 x2 + 5x – 28

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).

Example 2A: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. 1 3 (3x2 + 9x – 2) ÷ (x – ) Step 1 Find a. Then write the coefficients and a in the synthetic division format. 1 3 a = For (x – ), a = . 1 3 1 3 3 9 –2 Write the coefficients of 3x2 + 9x – 2.

Example 2A Continued Step 2 Bring down the first coefficient. Then multiply and add for each column. 1 3 3 9 –2 1 3 Draw a box around the remainder, 1 . 1 3 1 1 3 3 10 Step 3 Write the quotient. 3x + 10 + 1 3 x –

Example 2A Continued 3x + 10 + 1 3 x – Check Multiply (x – ) (x – ) 1 3 3x + 10 + x – = 3x2 + 9x – 2

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 0 5 –1 –2 Use 0 for the coefficient of x2.

Example 2B Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. –2 3 –1 0 5 –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.

Check It Out! Example 2a 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.

Check It Out! Example 2a Continued 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.

Check It Out! Example 2b Divide using synthetic division. (x2 – 3x – 18) ÷ (x – 6) Step 1 Find a. a = 6 For (x – 6), a = 6. Step 2 Write the coefficients and a in the synthetic division format. 6 1 –3 –18 Write the coefficients of x2 – 3x – 18.

Check It Out! Example 2b Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. 6 1 –3 –18 There is no remainder. 6 18 1 3 Step 4 Write the quotient. x + 3

Check It Out! Example 3 Write an expression for the length of a rectangle with width y – 9 and area y2 – 14y + 45. 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) = Substitute. 9 1 –14 45 Use synthetic division. 9 –45 1 –5 The length of the rectangle can be represented by l(x)= y – 5.

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. 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). 2x2 – 7x + 1