Synthetic division is a shorthand method of dividing

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 as a coefficient for any missing terms, and the divisor must be in the form (x – a).

Example 2 A: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (3 x 2 + 9 x – 2) ÷ (x – 1 ) 3 Step 1 Find a. Then write the coefficients and a in the synthetic division format. a= 1 3 3 For (x – 1 ), a = 31. 3 9 – 2 Write the coefficients of 3 x 2 + 9 x – 2.

Example 2 A Continued Step 2 Bring down the first coefficient. Then multiply and add for each column. 1 3 3 9 – 2 1 31 3 3 10 1 1 3 1 Draw a box around the remainder, 1 3. Step 3 Write the quotient. 11 3 3 x + 10 + 1 x– 3

Example 2 A Continued 1 3 Check Multiply (x – 1 ) 3 x + 10 + 3 1 x– 3 11 1 1 3 (x – 1 ) (x – ) 3 x 3 + 10 3 + 3 1 x– 3 = 3 x 2 + 9 x – 2 1

Example 2 B: Using Synthetic Division to Divide by a Linear Binomial Divide using synthetic division. (3 x 4 – x 3 + 5 x – 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. – 2 3 – 1 0 5 – 1 Use 0 for the coefficient of x 2.

Example 2 B Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. – 2 3 – 1 0 5 – 1 – 6 14 – 28 46 Draw a box around the remainder, 45. 3 – 7 14 – 23 45 Step 4 Write the quotient. 3 x 3 – 7 x 2 + 14 x – 23 + 45 x+2 Write the remainder over the divisor.

Check It Out! Example 2 a Divide using synthetic division. (6 x 2 – 5 x – 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 6 x 2 – 5 x – 6.

Check It Out! Example 2 a Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. – 3 6 – 5 – 6 – 18 69 6 Draw a box around the remainder, 63. – 23 63 Step 4 Write the quotient. 6 x – 23 + 63 x+3 Write the remainder over the divisor.

Check It Out! Example 2 b Divide using synthetic division. (x 2 – 3 x – 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 x 2 – 3 x – 18.

Check It Out! Example 2 b Continued Step 3 Bring down the first coefficient. Then multiply and add for each column. 6 1 1 – 3 – 18 6 18 3 0 There is no remainder. Step 4 Write the quotient. x+3