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

2. Copyright © Cengage Learning. All rights reserved.
Section
Dividing Polynomials by Monomials
4.7
Copyright © Cengage Learning. All rights reserved.

3. Objectives 1
Divide a monomial by a monomial. Divide a polynomial by a monomial. Solve a formula for a specified variable. 2 3

4. Divide a monomial by a monomial1.

5. Divide a monomial by a monomial
We have seen that dividing by a number is equivalent to multiplying by its reciprocal. For example, dividing the number 8 by 2 gives the same answer as multiplying 8 by and

6. Divide a monomial by a monomial
In general, the following is true. Division Recall that to simplify a fraction, we write both its numerator and its denominator as the product of several factors and then divide out all common factors.

7. Divide a monomial by a monomial
For example,

8. Divide a monomial by a monomial
We can use the same method to simplify algebraic fractions that contain variables. We must assume, however, that no variable is 0.
Factor: p2 = p  p, 6 = 2  3, and q3 = q  q  q.
Divide out the common factors of 3, p, and q.

9. Divide a monomial by a monomial
To divide monomials, we can either use the previous method or use the rules of exponents. Comment In all examples and exercises in this section, we will assume that no variables are 0 to avoid the possibility of division by 0. Recall that division by 0 is undefined.

10. Example
Simplify. Solution: Using Fractions Using the Rules of Exponents

11. Example – Solution
cont’d
Using Fractions Using the Rules of Exponents

12. Divide a polynomial by a monomial2.

13. Divide a polynomial by a monomial
We saw that Since this is true, we also have This suggests that, to divide a polynomial by a monomial, we can divide each term of the polynomial in the numerator by the monomial in the denominator.

14. Example Simplify: Solution:
Divide each term in the numerator by the monomial.
Simplify each fraction.

15. Solve a formula for a specified variable3.

16. Solve a formula for a specified variable
The cross-sectional area of the trapezoidal drainage ditch shown in Figure 4-7 is given by the formula where B and b are its bases and h is its height. To solve the formula for b, we proceed as follows.
Figure 4-7

17. Solve a formula for a specified variable
A = h(B + b)
2(A) = 2  h(B + b)
2A = h(B + b)
2A = hB + hb
2A – hB = hB – hB + hb
2A – hB = hb
Multiply both sides by 2 to clear fractions.
Simplify.
Use the distributive property to remove parentheses.
Subtract hB from both sides.
Combine like terms.
Divide both sides by h.

18. Example
Another student worked the previous problem in a different way and got a result of Is this also correct? Solution: To show that this result is correct, we must show that

19. Example – Solution
cont’d
We can do this by dividing 2A – hB by h. The results are the same.
Divide each term in the numerator by the monomial.
Simplify: