1. Copyright © Cengage Learning. All rights reserved. Polynomials 4

2. Copyright © Cengage Learning. All rights reserved. Section 4.8 Dividing Polynomials by Polynomials

3. 3 Objectives Divide a polynomial by a binomial. Divide a polynomial by a binomial by first writing exponents in descending order. Divide a polynomial with one or more missing terms by a binomial. 1 1 2 2 3 3

4. 4 Divide a polynomial by a binomial 1.

5. 5 Divide a polynomial by a binomial To divide one polynomial by another, we use a method similar to long division in arithmetic. Recall that the parts of a division problem are defined as Recall that division by zero is undefined. Therefore, the divisor cannot be 0. We must exclude any value of the variable that will result in a divisor of zero.

6. 6 Example 1 Divide (x 2 + 5x + 6) by (x + 2) Assume no division by 0. Solution: Here the divisor is x + 2 and the dividend is x 2 + 5x + 6. We proceed as follows: Step1: Step 2: How many times does x divide x 2 ? Write x above the division symbol. Multiply each item in the divisor by x. Write the product under x 2 + 5x and draw a line.

7. 7 Example 1 – Solution Step 3: Step 4: cont’d Subtract (x 2 + 2x) from (x 2 + 5x) by adding the negative of (x 2 + 2x) to (x 2 + 5x). Bring down the 6. How many times does x divide 3x? = +3 Write +3 above the division symbol.

8. 8 Example 1 – Solution Step 5: Step 6: cont’d Multiply each term in the divisor by 3. Write the product under the 3x + 6 and draw a line. Subtract (3x + 6) from (3x + 6) by adding the negative of (3x + 6).

9. 9 Example 1 – Solution The quotient is x + 3, and the remainder is 0. Step 7: Check by verifying that x + 2 times x + 3 is x 2 + 5x + 6. (x + 2)(x + 3) = x 2 + 3x + 2x + 6 = x 2 + 5x + 6 cont’d

10. 10 Divide a polynomial by a binomial by first writing exponents in descending order 2.

11. 11 Divide a polynomial by a binomial by first writing exponents in descending order The division method works best when exponents of the terms in the divisor and the dividend are written in descending order. This means that the term involving the highest power of x appears first, the term involving the second-highest power of x appears second, and so on. For example, the terms in 3x 3 + 2x 2 – 7x + 5 have their exponents written in descending order. 5 = 5x 0

12. 12 Divide a polynomial by a binomial by first writing exponents in descending order If the powers in the dividend or divisor are not in descending order, we can use the commutative property of addition to write them that way.

13. 13 Example 4 Divide: Assume no division by 0. Solution: We write the dividend so that the exponents are in descending order and divide.

14. 14 Example 4 – Solution Check: (x + 3)(2x 2 – 2x + 4) = 2x 3 – 2x 2 + 4x + 6x 2 – 6x + 12 = 2x 3 + 4x 2 – 2x + 12 cont’d

15. 15 Divide a polynomial with one or more missing terms by a binomial 3.

16. 16 Divide a polynomial with one or more missing terms by a binomial When we write the terms of a dividend in descending powers of x, we may notice that some powers of x are missing. For example, if the dividend of 3x 4 – 7x 2 – 3x + 15, the term involving x 3 is missing. When this happens, we should either write the term with a coefficient of 0 or leave a blank space for it. In this case, we would write the dividend as 3x 4 + 0x 3 – 7x 2 – 3x + 15 or 3x 4 – 7x 2 – 3x + 15

17. 17 Example Divide: Assume no division by 0. Solution: Since x 2 – 4 does not have a term involving x, we must either include the term 0x or leave a space for it.

18. 18 Example – Solution Check: (x + 2)(x – 2) = x 2 – 2x + 2x – 4 = x 2 – 4 cont’d