Dividing Polynomials - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST - Easy Algebra Step-by-Step

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 11. Dividing Polynomials

This chapter presents a discussion of division of polynomials. Division of polynomials is analogous to division of real numbers. In algebra, you indicate division using the fraction bar. For example, Image, x ≠ 0, indicates Image divided by –4x. Because division by 0 is undefined, you must exclude values for the variable or variables that would make the divisor 0. For convenience, you can assume such values are excluded as you work through the problems in this chapter.

Dividing a Polynomial by a Monomial

Customarily, a division problem is a dividend divided by a divisor. When you do the division, you get a quotient and a remainder. You express the relationship between these quantities as

Image

Image Dividing a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.


Be sure to note that the remainder is the numerator of the expression Image.


To avoid sign errors when you are doing division of polynomials, keep asymbol with the number that follows it. You likely will need to properly insert a + symbol when you do this.


Sign errors are a major reason for mistakes in division of polynomials.


You will see this tactic illustrated in the following problem.

Problem Find the quotient and remainder.

a. Image

b. Image

c. Image

d. Image

e. Image

Solution

a. Image

Image Step 1. Divide each term of the polynomial by the monomial.

Image

Step 2. State the quotient and remainder.

The quotient is Image and the remainder is 0.

b. Image

Image Step 1. Divide each term of the polynomial by the monomial.

Image

Step 2. State the quotient and remainder.

The quotient is 4x3 – 2x and the remainder is 0.

c. Image

Image Step 1. Divide each term of the polynomial by the monomial.

Image

Step 2. State the quotient and remainder.

The quotient is x3 – 2x2y2 + 4y3 and the remainder is 0.

d. Image

Image Step 1. Divide each term of the polynomial by the monomial.

Image

Image

Step 2. State the quotient and remainder.

The quotient is 3 and the remainder is 1.

e. Image

Image Step 1. Divide 16x5y2 by 16x5y2

Image

Step 2. State the quotient and remainder.

The quotient is 1 and the remainder is 0.

Dividing a Polynomial by a Polynomial

When you divide two polynomials, and the divisor is not a monomial, you use long division. The procedure is very similar to the long division algorithm of arithmetic. The steps are illustrated in the following problem.

Problem Find the quotient and remainder.

a. Image

b. Image

c. Image

Solution

a. Image

Image Step 1. Using the long division symbol Image, arrange the terms of both the dividend and the divisor in descending powers of the variable x.

Image

Step 2. Divide the first term of the dividend by the first term of the divisor and write the answer as the first term of the quotient.

Image

Step 3. Multiply 2x – 1 by 2x2 and enter the product under the dividend.

Image

Step 4. Subtract 4x3-2x2 from the dividend, being sure to mentally change the signs of both 4x3 and –2x2.

Image


In long division of polynomials, making sign errors when subtracting is the most common mistake.


Step 5. Bring down 8x, the next term of the dividend, and repeat steps 2–4.

Image

Step 6. Bring down 1, the last term of the dividend, and repeat steps 2–4.

Image

Step 7. State the quotient and remainder.

The quotient is Image and the remainder is 4.

b. Image

Image Step 1. Using the long division symbol Image, arrange the terms of both the dividend and the divisor in descending powers of the variable x. Insert zeros as placeholders for missing powers of x.

Image


Imagex2 + 4. Avoid this common error.


Step 2. Divide the first term of the dividend by the first term of the divisor and write the answer as the first term of the quotient.

Image

Step 3. Multiply x – 2 by x2 and enter the product under the dividend.

Image

Step 4. Subtract x3 – 2x2 from the dividend, being sure to mentally change the signs of both x3 and –2x2.

Image

Step 5. Bring down 0, the next term of the dividend, and repeat steps 2–4.

Image

Step 6. Bring down –8, the last term of the dividend, and repeat steps 2–4.

Image

Step 7. State the quotient and remainder.

The quotient is Image and the remainder is 0.

c. Image

Image Step 1. Using the long division symbol Image, arrange the terms of both the dividend and the divisor in descending powers of the variable x.

Image

Step 2. Divide the first term of the dividend by the first term of the divisor and write the answer as the first term of the quotient.

Image

Step 3. Multiply x + 3 by x and enter the product under the dividend.

Image

Step 4. Subtract x2 + 3x from the dividend, being sure to mentally change the signs of both x2 and 3x.

Image

Step 5. Bring down –4, the next term of the dividend, and repeat steps 2–4.

Image

Step 6. State the quotient and remainder.

The quotient is x – 2 and the remainder is 2.

Image Exercise 11

Find the quotient and remainder.

1. Image

2. Image

3. Image

4. Image

5. Image

6. Image

7. Image

8. Image

9. Image

10. Image