## 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, , *x* ≠ 0, indicates divided by –4*x*. 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

** 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 .

To avoid sign errors when you are doing division of polynomials, *keep a* – *symbol 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**.

**b**.

**c**.

**d**.

**e**.

**Solution**

**a**.

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

*Step 2*. State the quotient and remainder.

The quotient is and the remainder is 0.

**b**.

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

*Step 2*. State the quotient and remainder.

The quotient is 4*x*^{3} – 2*x* and the remainder is 0.

**c**.

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

*Step 2*. State the quotient and remainder.

The quotient is x^{3} – 2x^{2}y^{2} + 4y^{3} and the remainder is 0.

**d**.

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

*Step 2*. State the quotient and remainder.

The quotient is 3 and the remainder is 1.

**e**.

*Step 1*. Divide 16x^{5}y^{2} by 16x^{5}y^{2}

*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**.

**b**.

**c**.

**Solution**

**a**.

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

*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.

*Step 3*. Multiply 2*x* – 1 by 2*x*^{2} and enter the product under the dividend.

*Step 4*. Subtract 4*x*^{3}-2*x*^{2} from the dividend, being sure to mentally change the signs of *both* 4*x*^{3} and –2x^{2}.

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

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

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

*Step 7*. State the quotient and remainder.

The quotient is and the remainder is 4.

**b**.

*Step 1*. Using the long division symbol , 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*.

≠*x*^{2} + 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.

*Step 3*. Multiply *x* – 2 by *x*^{2} and enter the product under the dividend.

*Step 4*. Subtract *x*^{3} – 2*x*^{2} from the dividend, being sure to mentally change the signs of *both x*^{3} and –2*x*^{2}.

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

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

*Step 7*. State the quotient and remainder.

The quotient is and the remainder is 0.

**c**.

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

*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.

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

*Step 4*. Subtract *x*^{2} + 3*x* from the dividend, being sure to mentally change the signs of *both x*^{2} and 3*x*.

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

*Step 6*. State the quotient and remainder.

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

**Exercise 11**

*Find the quotient and remainder**.*

__1__.

__2__.

__3__.

__4__.

__5__.

__6__.

__7__.

__8__.

__9__.

__10__.