Dividing Polynomials Long division of polynomials is similar

Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. When you divide two polynomials you can check the answer using the following: dividend = (quotient • divisor) + remainder The result is written in the form: quotient + Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1

Long Division

Example: Divide x 2 + 3 x – 2 by x – 1 and check the answer. x + 2 1. 2. x 2 + x 3. 2 x – 2 2 x + 2 4. – 4 5. remainder Answer: x + 2 + Check: (x + 2) 6. – 4 (x + 1) + (– 4) = x 2 + 3 x – 2 quotient divisor remainder Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 dividend correct

Example: Divide 4 x + 2 x 3 – 1 by 2 x – 2 and check the answer. x 2 + x + 3 Write the terms of the dividend in descending order. 2 x 3 – 2 x 2 Since there is no x 2 term in the dividend, add 0 x 2 as a placeholder. + 4 x 2 x 2 – 2 x 1. 6 x – 1 3. 6 x – 6 5 Answer: x 2 +x+3 4. 5 6. 7. Check: (x 2 + x + 3)(2 x – 2) + 5 = 4 x + 2 x 3 – 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2. 9. 4 8.

Example: Divide x 2 – 5 x + 6 by x – 2. x – 3 x 2 – 2 x – 3 x + 6 0 Answer: x – 3 with no remainder. Check: (x – 2)(x – 3) = x 2 – 5 x + 6 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

Synthetic Division Use synthetic division to find (x 3 – 4 x 2 + 6 x – 4) ÷ (x – 2). Step 1 Write the terms of the x 3 – 4 x 2 + 6 x – 4 dividend so that the degrees of the terms are in – 6 descending order. Then write 1 – 4 4 just the coefficients as shown. Step 2 Write the constant r of 1 – 4 6 the divisor x – r to the – 4 left. In this case, r = 2. 1 Bring the first coefficient, 1, down as shown.

Synthetic Division Step 3 Multiply the first coefficient by r : 1 ● 2 = 2. Write the product under the second coefficient. Then add the product and the second coefficient. Step 4 Multiply the sum, – 2, by r : 2(– 2) = – 4. Write the product under the next coefficient and add: 6 + (– 4) = 2. 1 6 2 – 4 1 – 2 1 – 4 6 – 4 2 – 4 1 – 2 2

Synthetic Division Step 5 Multiply the sum, 2, by r : 2(2) = 4. Write the product under the next coefficient and add: – 4 + 4 = 0. The remainder is 0. 1 – 4 6 – 4 4 2 – 4 1 – 2 2 0 The numbers along the bottom are the coefficients of the quotient. Start with the power of x that is one less than the degree of the dividend. Answer: The quotient is x 2 – 2 x + 2.

Example Use synthetic division to find (x 2 + 8 x + 7) ÷ (x + 1).

Divide by x + 3