540 Chapter 5 and 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 , we write one 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.

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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 . 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 . 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 , 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 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.

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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

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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 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.

Write the divisor as x − c and place c in the synthetic division in the divisor box.

Bring down the first to the third row.

Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.

Add the second column, putting the result in the third row.

Multiply that result by the divisor and place the result in the second row under the third coefficient.

Add the third column, putting the result in the third row.

Multiply that result by the divisor and place the result in the third row under the third coefficient. 546 Chapter 5 Polynomials and Polynomial Functions

Add the final column, putting the result in the third row.

The quotient is 2x2 − 1x + 3 and the remainder is 2.

The division is complete. The numbers in the third row give us the result. The 2 −1 3 are the coefficients of the quotient. The quotient is 2x2 − 1x + 3. The 2 in the box in the third row is the remainder. Check:

(quotient)(divisor) + remainder = dividend ⎛ 2 ⎞ ? 3 2 ⎝2x − 1x + 3⎠(x + 2) + 2 = 2x + 3x + x + 8 2x3 − x2 + 3x + 4x2 − 2x + 6 + 2 =? 2x3 + 3x2 + x + 8 2x3 + 3x2 + x + 8 = 2x3 + 3x2 + x + 8 ✓

TRY IT : : 5.83

Use synthetic division to find the quotient and remainder when 3x3 + 10x2 + 6x − 2 is divided by x + 2.

TRY IT : : 5.84

Use synthetic division to find the quotient and remainder when 4x3 + 5x2 − 5x + 3 is divided by x + 2.

In the next example, we will do all the steps together.

EXAMPLE 5.43

Use synthetic division to find the quotient and remainder when x4 − 16x2 + 3x + 12 is divided by x + 4. Solution The polynomial x4 − 16x2 + 3x + 12 has its term in order with descending degree but we notice there is no x3 term. We will add a 0 as a placeholder for the x3 term. In x − c form, the divisor is x − (−4).

We divided a 4th degree polynomial by a 1st degree polynomial so the quotient will be a 3rd degree polynomial. Reading from the third row, the quotient has the coefficients 1 −4 0 3, which is x3 − 4x2 + 3. The remainder is 0.

TRY IT : : 5.85

Use synthetic division to find the quotient and remainder when x4 − 16x2 + 5x + 20 is divided by x + 4.

TRY IT : : 5.86

Use synthetic division to find the quotient and remainder when x4 − 9x2 + 2x + 6 is divided by x + 3.

Divide Polynomial Functions Just as polynomials can be divided, polynomial functions can also be divided.

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Division of Polynomial Functions

For functions f (x) and g(x), where g(x) ≠ 0, ⎛ f ⎞ f (x) (x) = ⎝g⎠ g(x)

EXAMPLE 5.44

2 ⎛ f ⎞ ⎛ f ⎞ For functions f (x) = x − 5x − 14 and g(x) = x + 2, find: ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(−4).

Solution ⓐ

⎛ f ⎞ x2 x Substitute for f (x) and g(x). (x) = − 5 − 14 ⎝g⎠ x + 2 ⎛ f ⎞ Divide the polynomials. ⎝g⎠(x) = x − 7

⎛ f ⎞ ⎛ f ⎞ ⓑ In part ⓐ we found ⎝g⎠(x) and now are asked to find ⎝g⎠(−4). ⎛ f ⎞ ⎝g⎠(x) = x − 7 ⎛ f ⎞ ⎛ f ⎞ To fin ⎝g⎠(−4), substitute x = −4. ⎝g⎠(−4) = −4 − 7

⎛ f ⎞ ⎝g⎠(−4) = −11

TRY IT : : 5.87 2 ⎛ f ⎞ ⎛ f ⎞ For functions f (x) = x − 5x − 24 and g(x) = x + 3, find ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(−3).

TRY IT : : 5.88 2 ⎛ f ⎞ ⎛ f ⎞ For functions f (x) = x − 5x − 36 and g(x) = x + 4, find ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(−5).

Use the Remainder and Factor Theorem Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as x − c, the value of the function at c, f (c), is the same as the remainder from the division problem. 548 Chapter 5 Polynomials and Polynomial Functions

Dividend Divisor x − c Remainder Function f(c)

x4 − x2 + 5x − 6 x − (−2) −4 f (x) = x4 − x2 + 5x − 6 −4

3x3 − 2x2 − 10x + 8 x − 2 4 f (x) = 3x3 − 2x2 − 10x + 8 4

x4 − 16x2 + 3x + 15 x − (−4) 3 f (x) = x4 − 16x2 + 3x + 15 3

Table 5.44

To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend f (x), we multiply the quotient, q(x) times the divisor, x − c, and add the remainder, r.

If we evaluate this at c, we get:

This leads us to the Remainder Theorem.

Remainder Theorem

If the polynomial function f (x) is divided by x − c, then the remainder is f (c).

EXAMPLE 5.45

Use the Remainder Theorem to find the remainder when f (x) = x3 + 3x + 19 is divided by x + 2. Solution To use the Remainder Theorem, we must use the divisor in the x − c form. We can write the divisor x + 2 as x − (−2). So, our c is −2. To find the remainder, we evaluate f (c) which is f (−2).

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To evaluate f (−2), substitute x = −2.

Simplify.

The remainder is 5 when f (x) = x3 + 3x + 19 is divided by x + 2.

Check: Use synthetic division to check.

The remainder is 5.

TRY IT : : 5.89

Use the Remainder Theorem to find the remainder when f (x) = x3 + 4x + 15 is divided by x + 2.

TRY IT : : 5.90

Use the Remainder Theorem to find the remainder when f (x) = x3 − 7x + 12 is divided by x + 3.

When we divided 8a3 + 27 by 2a + 3 in Example 5.41 the result was 4a2 − 6a + 9. To check our work, we multiply 4a2 − 6a + 9 by 2a + 3 to get 8a3 + 27 . ⎛ 2 ⎞ 3 ⎝4a − 6a + 9⎠(2a + 3) = 8a + 27

Written this way, we can see that 4a2 − 6a + 9 and 2a + 3 are factors of 8a3 + 27. When we did the division, the remainder was zero. Whenever a divisor, x − c, divides a polynomial function, f (x), and resulting in a remainder of zero, we say x − c is a factor of f (x). The reverse is also true. If x − c is a factor of f (x) then x − c will divide the polynomial function resulting in a remainder of zero. We will state this in the Factor Theorem.

Factor Theorem

For any polynomial function f (x), • if x − c is a factor of f (x), then f (c) = 0 • if f (c) = 0, then x − c is a factor of f (x)

EXAMPLE 5.46

Use the Remainder Theorem to determine if x − 4 is a factor of f (x) = x3 − 64. 550 Chapter 5 Polynomials and Polynomial Functions

Solution The Factor Theorem tells us that x − 4 is a factor of f (x) = x3 − 64 if f (4) = 0. f (x) = x3 − 64 To evaluate f (4) substitute x = 4. f (4) = 43 − 64 Simplify. f (4) = 64 − 64 Subtract. f (4) = 0

Since f (4) = 0, x − 4 is a factor of f (x) = x3 − 64.

TRY IT : : 5.91 Use the Factor Theorem to determine if x − 5 is a factor of f (x) = x3 − 125.

TRY IT : : 5.92 Use the Factor Theorem to determine if x − 6 is a factor of f (x) = x3 − 216.

MEDIA : : Access these online resources for additional instruction and practice with dividing polynomials. • Dividing a Polynomial by a Binomial (https://openstax.org/l/37Polybybinom) • Synthetic Division & Remainder Theorem (https://openstax.org/l/37SynDivision)

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5.4 EXERCISES Practice Makes Perfect

Divide Monomials In the following exercises, divide the monomials. 4 9 ⎛ 4 9⎞ 8 4 ⎛ 5 9⎞ 4 8 288. 15r s ÷ ⎝15r s ⎠ 289. 20m n ÷ ⎝30m n ⎠ 290. 18a b −27a9 b5

x5 y9 ⎛ m5 n4⎞⎛ m3 n6⎞ ⎛ p4 q7⎞⎛ p3 q8⎞ 291. 45 ⎝10 ⎠⎝5 ⎠ ⎝−18 ⎠⎝−6 ⎠ 8 6 292. 293. −60x y 25m7 n5 −36p12 q10

⎛ 4 3⎞⎛ 5⎞ ⎛ 2 5⎞⎛ 3 ⎞ ⎝6a b ⎠⎝4ab ⎠ ⎝4u v ⎠⎝15u v⎠ 294. 295. ⎛ 2 ⎞⎛ 3 ⎞ ⎛ 3 ⎞⎛ 4 ⎞ ⎝12a b⎠⎝a b⎠ ⎝12u v⎠⎝u v⎠

Divide a Polynomial by a Monomial In the following exercises, divide each polynomial by the monomial. ⎛ 4 3⎞ ⎛ 3 2⎞ ⎛ 4 3⎞ ⎛ 2⎞ 296. ⎝9n + 6n ⎠ ÷ 3n 297. ⎝8x + 6x ⎠ ÷ 2x 298. ⎝63m − 42m ⎠ ÷ ⎝−7m ⎠

⎛ 4 3⎞ ⎛ 2⎞ 300. 301. 299. ⎝48y − 24y ⎠ ÷ ⎝−8y ⎠ 66x3 y2 − 110x2 y3 − 44x4 y3 72r5 s2 + 132r4 s3 − 96r3 s5 2 2 11x2 y2 12r s

x2 x y2 y 302. 10 + 5 − 4 303. 20 + 12 − 1 −5x −4y

Divide Polynomials using Long Division In the following exercises, divide each polynomial by the binomial.

⎛ 2 ⎞ ⎛ ⎞ ⎛ 2 ⎞ 306. 304. ⎝y + 7y + 12⎠ ÷ ⎝y + 3⎠ 305. ⎝a − 2a − 35⎠ ÷ (a + 5) ⎛ 2 ⎞ ⎝6m − 19m − 20⎠ ÷ (m − 4)

⎛ 2 ⎞ ⎛ 2 ⎞ ⎛ ⎞ ⎛ 2 ⎞ ⎛ ⎞ 307. ⎝4x − 17x − 15⎠ ÷ (x − 5) 308. ⎝q + 2q + 20⎠ ÷ ⎝q + 6⎠ 309. ⎝p + 11p + 16⎠ ÷ ⎝p + 8⎠

⎛ 3 2 ⎞ ⎛ 3 ⎞ ⎛ 3 ⎞ 310. ⎝3b + b + 4⎠ ÷ (b + 1) 311. ⎝2n − 10n + 28⎠ ÷ (n + 3) 312. ⎝z + 1⎠ ÷ (z + 1)

⎛ 3 ⎞ ⎛ 3 ⎞ ⎛ 3 ⎞ ⎛ ⎞ 313. ⎝m + 1000⎠ ÷ (m + 10) 314. ⎝64x − 27⎠ ÷ (4x − 3) 315. ⎝125y − 64⎠ ÷ ⎝5y − 4⎠

Divide Polynomials using Synthetic Division In the following exercises, use synthetic Division to find the quotient and remainder. 316. x3 − 6x2 + 5x + 14 is divided by x + 1 317. x3 − 3x2 − 4x + 12 is divided by x + 2

318. 2x3 − 11x2 + 11x + 12 is divided by x − 3 319. 2x3 − 11x2 + 16x − 12 is divided by x − 4

320. x4 + 13x2 + 13x + 3 is divided by x + 3 321. x4 + x2 + 6x − 10 is divided by x + 2 552 Chapter 5 Polynomials and Polynomial Functions

322. 2x4 − 9x3 + 5x2 − 3x − 6 is divided by x − 4 323. 3x4 − 11x3 + 2x2 + 10x + 6 is divided by x − 3

Divide Polynomial Functions In the following exercises, divide. 324. For functions f (x) = x2 − 13x + 36 and 325. For functions f (x) = x2 − 15x + 45 and ⎛ f ⎞ ⎛ f ⎞ ⎛ f ⎞ ⎛ f ⎞ g(x) = x − 4, find ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(−1) g(x) = x − 9, find ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(−5)

326. For functions f (x) = x3 + x2 − 7x + 2 and 327. For functions f (x) = x3 + 2x2 − 19x + 12 and ⎛ f ⎞ ⎛ f ⎞ ⎛ f ⎞ ⎛ f ⎞ g(x) = x − 2, find ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(2) g(x) = x − 3, find ⓐ ⎝g⎠(x) ⓑ ⎝g⎠(0)

328. For functions f (x) = x2 − 5x + 2 and 329. For functions f (x) = x2 + 4x − 3 and

2 ⎛ ⎞ ⎛ ⎞ 2 ⎛ ⎞ ⎛ ⎞ g(x) = x − 3x − 1, find ⓐ ⎝ f · g⎠(x) ⓑ ⎝ f · g⎠(−1) g(x) = x + 2x + 4, find ⓐ ⎝ f · g⎠(x) ⓑ ⎝ f · g⎠(1)

Use the Remainder and Factor Theorem In the following exercises, use the Remainder Theorem to find the remainder. 330. f (x) = x3 − 8x + 7 is divided by x + 3 331. f (x) = x3 − 4x − 9 is divided by x + 2

332. f (x) = 2x3 − 6x − 24 divided by x − 3 333. f (x) = 7x2 − 5x − 8 divided by x − 1

In the following exercises, use the Factor Theorem to determine if x − c is a factor of the polynomial function. 334. Determine whether x + 3 a factor of 335. Determine whether x + 4 a factor of x3 + 8x2 + 21x + 18 x3 + x2 − 14x + 8

336. Determine whether x − 2 a factor of 337. Determine whether x − 3 a factor of x3 − 7x2 + 7x − 6 x3 − 7x2 + 11x + 3

Writing Exercises

338. James divides 48y + 6 by 6 this way: 2 339. Divide 10x + x − 12 and explain with words 48y + 6 2x = 48y. What is wrong with his reasoning? 6 how you get each term of the quotient.

340. Explain when you can use synthetic division. 341. In your own words, write the steps for synthetic division for x2 + 5x + 6 divided by x − 2.

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Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?