1. § 6.4 Division of Polynomials

2. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.4 Division of Polynomials In this section we will look at dividing by a monomial and dividing by a polynomial containing more than one term. Division of a polynomial by a monomial is relatively easy – you just divide each term of the polynomial by the monomial. The number of separate divisions you will have is the number of terms in the polynomial. The second case, that of dividing a polynomial by a polynomial having more than one term, is more difficult. That case requires a process of long division.

3. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.4 Division of Polynomials Dividing a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. 1 st case

4. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.4 Division of PolynomialsEXAMPLE Divide: SOLUTION Express the division in a vertical format. Divide each term of the polynomial by the monomial. Note the 3 separate quotients. Simplify each quotient.

5. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.4 Division of Polynomials Now, we will consider the harder problem – that of dividing a polynomial by a polynomial having more than one term. The four steps you use in long division of whole numbers – divide, multiply, subtract, bring down the next term – form the same repetitive procedure for polynomial long division. Carefully consider and try to remember the four terms illustrated on the next slide. These terms are: quotient, divisor, dividend, and remainder.

6. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.4 Division of Polynomials DIVISORQUOTIENT EXAMPLE REMAINDER DIVIDEND 2 nd case

7. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.4 Division of PolynomialsEXAMPLE Divide: SOLUTION Arrange the terms of the dividend,, and the divisor, (x + 2), in descending powers of x. Divide (the first term in the dividend) by x (the first term in the divisor). Align like terms.

8. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.4 Division of Polynomials Multiply each term in the divisor (x + 2) by, aligning terms of the product under like terms in the dividend. Subtract from by changing the sign of each term in the lower expression and then adding. CONTINUED

9. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.4 Division of Polynomials Bring down 7x from the original dividend and add algebraically to form a new dividend. Find the second term of the quotient. Divide the first term of by x, the first term of the divisor. CONTINUED

10. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.4 Division of Polynomials Multiply the divisor (x + 2) by 3x, aligning under like terms in the new dividend. Then subtract. CONTINUED

11. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.4 Division of Polynomials Bring down 2 from the original dividend and add algebraically to form a new dividend. Find the third term of the quotient, 1. Divide the first term of x + 2 by x, the first term of the divisor. Multiply the divisor by 1, aligning under like terms in the new dividend. Then subtract to obtain the remainder of 0. CONTINUED

12. Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.4 Division of Polynomials The quotient is and the remainder is 0. We will not list a remainder of 0 in the answer. Thus, CONTINUED

13. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.4 Long Division of Polynomials 1) Arrange the terms of both the dividend and the divisor in descending powers of any variable. 2) Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient. 3) Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. 4) Subtract the product from the dividend. 5) Bring down the next term in the original dividend and write it next to the remainder to form a new dividend. 6) Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. This will occur when the degree of the remainder (the highest exponent on a variable in the remainder) is less than the degree of the divisor.

14. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.4 Long Division of PolynomialsEXAMPLE Divide: SOLUTION We write the dividend,, as to keep all like terms aligned. For the same reason, we write the divisor,, as

15. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 6.4 Long Division of PolynomialsCONTINUED

16. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 6.4 Long Division of PolynomialsCONTINUED The division process is finished because the degree of -2x, which is 1, is less than the degree of the divisor, which is 2. The answer is

17. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 6.4 Long Division of Polynomials Important to Remember: To divide by a polynomial containing more than one term, use long division. If necessary, arrange the dividend in descending powers of the variable. Do the same with the divisor. If a power of a variable is missing in the dividend, add that term using a coefficient of 0. Repeat the four steps of the long-division process – divide, multiply, subtract, bring down the next term – until the degree of the remainder is less than the degree of the divisor. When the degree of the remainder is less than the degree of the divisor – you know you are done!