540 Chapter 5 Polynomials and Polynomial Functions 5.4 Dividing Polynomials Learning Objectives By the end of this section, you will be able to: Dividing monomials Dividing a polynomial by a monomial Dividing polynomials using long division Dividing polynomials using synthetic division Dividing polynomial functions Use the remainder and factor theorems Be Prepared! Before you get started, take this readiness quiz. 1. Add: 3 x d + d. If you missed this problem, review Example 1.28. 30xy3 2. Simplify: . 5xy If you missed this problem, review Example 1.25. 3. Combine like terms: 8a2 + 12a + 1 + 3a2 − 5a + 4. If you missed this problem, review Example 1.7. Dividing Monomials We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials. EXAMPLE 5.36 2 3 ⎛ 5⎞ Find the quotient: 54a b ÷ ⎝−6ab ⎠. Solution When we divide monomials with more than one variable, we write one fraction for each variable. 2 3 ⎛ 5⎞ 54a b ÷ ⎝−6ab ⎠ a2 b3 Rewrite as a fraction. 54 −6ab5 54 a2 b3 Use fraction multiplication. · a · −6 b5 Simplify and use the Quotient Property. −9 · a · 1 b2 a Multiply. −9 b2 TRY IT : : 5.71 7 3 ⎛ 12 4⎞ Find the quotient: −72a b ÷ ⎝8a b ⎠. TRY IT : : 5.72 8 3 ⎛ 12 2⎞ Find the quotient: −63c d ÷ ⎝7c d ⎠. Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step. This OpenStax book is available for free at http://cnx.org/content/col12119/1.3 Chapter 5 Polynomials and Polynomial Functions 541 EXAMPLE 5.37 14x7 y12 Find the quotient: . 21x11 y6 Solution Be very careful to simplify 14 by dividing out a common factor, and to simplify the variables by subtracting their 21 exponents. 14x7 y12 21x11 y6 2y6 Simplify and use the Quotient Property. 3x4 TRY IT : : 5.73 28x5 y14 Find the quotient: . 49x9 y12 TRY IT : : 5.74 m5 n11 Find the quotient: 30 . 48m10 n14 Divide a Polynomial by a Monomial Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial. The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start y y + 2 with an example to review fraction addition. The sum + 2 simplifies to . 5 5 5 y + 2 y Now we will do this in reverse to split a single fraction into separate fractions. For example, can be written + 2. 5 5 5 a + b a b This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where c ≠ 0, then c = c + c. We will use this to divide polynomials by monomials. Division of a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. EXAMPLE 5.38 ⎛ 3 2⎞ ⎛ ⎞ Find the quotient: ⎝18x y − 36xy ⎠ ÷ ⎝−3xy⎠. Solution ⎛ 3 2⎞ ⎛ ⎞ ⎝18x y − 36xy ⎠ ÷ ⎝−3xy⎠ 18x3 y − 36xy2 Rewrite as a fraction. −3xy 18x3 y 36xy2 Divide each term by the divisor. Be careful with the signs! − −3xy −3xy Simplify. −6x2 + 12y TRY IT : : 5.75 ⎛ 2 2⎞ Find the quotient: ⎝32a b − 16ab ⎠ ÷ (−8ab). 542 Chapter 5 Polynomials and Polynomial Functions TRY IT : : 5.76 ⎛ 8 4 6 5⎞ ⎛ 3 3⎞ Find the quotient: ⎝−48a b − 36a b ⎠ ÷ ⎝−6a b ⎠. Divide Polynomials Using Long Division Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25. We check division by multiplying the quotient by the divisor. If we did the division correctly, the product should equal the dividend. 35 · 25 875 ✓ Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above. EXAMPLE 5.39 ⎛ 2 ⎞ Find the quotient: ⎝x + 9x + 20⎠ ÷ (x + 5). Solution Write it as a long division problem. Be sure the dividend is in standard form. Divide x2 by x. It may help to ask yourself, “What do I need to multiply x by to get x2 ?” Put the answer, x, in the quotient over the x term. Multiply x times x + 5. Line up the like terms under the dividend. Subtract x2 + 5x from x2 + 9x. You may find it easier to change the signs and then add. Then bring down the last term, 20. Divide 4x by x. It may help to ask yourself, “What do I need to multiply x by to get 4x ?” Put the answer, 4 , in the quotient over the constant term. Multiply 4 times x + 5. Subtract 4x + 20 from 4x + 20. This OpenStax book is available for free at http://cnx.org/content/col12119/1.3 Chapter 5 Polynomials and Polynomial Functions 543 Check: Multiply the quotient by the divisor. (x + 4)(x + 5) You should get the dividend. x2 + 9x + 20 ✓ TRY IT : : 5.77 ⎛ 2 ⎞ ⎛ ⎞ Find the quotient: ⎝y + 10y + 21⎠ ÷ ⎝y + 3⎠. TRY IT : : 5.78 ⎛ 2 ⎞ Find the quotient: ⎝m + 9m + 20⎠ ÷ (m + 4). When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator. Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be x4 − x2 + 5x − 6. It is missing an x3 term. We will add in 0x3 as a placeholder. EXAMPLE 5.40 ⎛ 4 2 ⎞ Find the quotient: ⎝x − x + 5x − 6⎠ ÷ (x + 2). Solution Notice that there is no x3 term in the dividend. We will add 0x3 as a placeholder. Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms. Divide x4 by x. Put the answer, x3, in the quotient over the x3 term. Multiply x3 times x + 2. Line up the like terms. Subtract and then bring down the next term. Divide −2x3 by x. Put the answer, −2x2, in the quotient over the x2 term. Multiply −2x2 times x + 1. Line up the like terms Subtract and bring down the next term. Divide 3x2 by x. Put the answer, 3x, in the quotient over the x term. Multiply 3x times x + 1. Line up the like terms. Subtract and bring down the next term. 544 Chapter 5 Polynomials and Polynomial Functions Divide −x by x. Put the answer, −1, in the quotient over the constant term. Multiply −1 times x + 1. Line up the like terms. Change the signs, add. Write the remainder as a fraction with the divisor as the denominator. To check, multiply ⎛ ⎞ (x + 2) x3 − 2x2 + 3x − 1 − 4 . ⎝ x + 2⎠ The result should be x4 − x2 + 5x − 6. TRY IT : : 5.79 ⎛ 4 2 ⎞ Find the quotient: ⎝x − 7x + 7x + 6⎠ ÷ (x + 3). TRY IT : : 5.80 ⎛ 4 2 ⎞ Find the quotient: ⎝x − 11x − 7x − 6⎠ ÷ (x + 3). In the next example, we will divide by 2a − 3. As we divide, we will have to consider the constants as well as the variables. EXAMPLE 5.41 ⎛ 3 ⎞ Find the quotient: ⎝8a + 27⎠ ÷ (2a + 3). Solution This time we will show the division all in one step. We need to add two placeholders in order to divide. ⎛ 2 ⎞ To check, multiply (2a + 3)⎝4a − 6a + 9⎠. The result should be 8a3 + 27. TRY IT : : 5.81 ⎛ 3 ⎞ Find the quotient: ⎝x − 64⎠ ÷ (x − 4). TRY IT : : 5.82 ⎛ 3 ⎞ Find the quotient: ⎝125x − 8⎠ ÷ (5x − 2). Divide Polynomials using Synthetic Division As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in Example 5.39 and look for some patterns. We will use this as a basis This OpenStax book is available for free at http://cnx.org/content/col12119/1.3 Chapter 5 Polynomials and Polynomial Functions 545 for what is called synthetic division. The same problem in the synthetic division format is shown next. Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the x and x2 are removed. as well as the −x2 and −4x as they are opposite the term above. The first row of the synthetic division is the coefficients of the dividend. The −5 is the opposite of the 5 in the divisor. The second row of the synthetic division are the numbers shown in red in the division problem. The third row of the synthetic division are the numbers shown in blue in the division problem. Notice the quotient and remainder are shown in the third row. Synthetic division only works when the divisor is of the form x − c. The following example will explain the process. EXAMPLE 5.42 Use synthetic division to find the quotient and remainder when 2x3 + 3x2 + x + 8 is divided by x + 2. Solution Write the dividend with decreasing powers of x. Write the coefficients of the terms as the first row of the synthetic division.