Domains of Rational Expressions

In Chapter 4, you learned how to multiply a monomial and a polynomial. In this section, you will learn how to reverse that multiplication by finding the greatest common factor for the terms of a polynomial and then factoring the polynomial.

To factor an expression means to write the expression as a product. For example, if we start with 12 and write 12 = 4 ⋅ 3, we have factored 12. Both 4 and 3 are factors or divisors of 12. There are other factorizations of 12:

The one that is most useful to us is 12 = 22 ⋅ 3, because it expresses 12 as a product of prime numbers.

A positive integer larger than1 that has no positive integral factors other than itself and 1 is called a prime number.

The numbers 2, 3, 5, 7, 11, 13, 17, 19, and 23 are the first nine prime numbers. A positive integer larger than 1 that is not a prime is a composite number. The numbers 4, 6, 8, 9, 10, and 12 are the first six composite numbers. Every composite number is a product of prime numbers. The prime factorization for 12 is 22 ⋅ 3.

Factoring Out the Greatest Common Factor

Find the prime factorization for 36.

We start by writing 36 as a product of two integers:

The prime factorization for 36 is 22 ⋅ 32.

Now do Exercises 1–6

The prime factorization of 36can be found also with a factoring tree:

So 36 = 2 ⋅ 2 ⋅ 3 ⋅ 3.

For larger integers, it is better to use the method shown in Example 2 and to recall some divisibility rules. Even numbers are divisible by 2. If the sum of the digits of a number is divisible by 3, then the number is divisible by 3. Numbers that end in 0 or 5 are divisible by 5. Two-digit numbers with repeated digits (11, 22, 33, …) are divisible by 11.

Find the prime factorization for 420.

Start by dividing 420 by the smallest prime number that will divide into it evenly (without remainder). The smallest prime divisor of 420 is 2.

Now find the smallest prime that will divide evenly into the quotient, 210. The smallest prime divisor of 210 is 2. Continue this procedure, as follows, until the quotient is a prime number:

The product of all of the prime numbers in this procedure is 420:

So the prime factorization of 420 is 22 ⋅ 3 ⋅ 5 ⋅ 7. Note that it is not necessary to divide by the smallest prime divisor at each step. We get the same factorization if we divide by any prime divisor.

Now do Exercises 7–12

The fact that every composite number has a unique prime factori-z ation is known as the fundamental theorem of arithmetic.

Note that the division in Example 2 can be done also as follows:

The largest integer that is a factor of two or more integers is called the greatest common factor (GCF) of the integers. For example, 1, 2, 3, and 6 are common factors of 18 and 24. Because 6 is the largest, 6 is the GCF of 18 and 24. We can use prime factorizations to find the GCF. For example, to find the GCF of 8 and 12, we first factor 8 and 12:

We see that the factor 2 appears twice in both 8 and 12. So 22, or 4, is the GCF of 8 and 12. Notice that 2 is a factor in both 23 and 22 ⋅ 3 and that 22 is the smallest power of 2 in these factorizations. In general, we can use the following strategy to find the GCF.

1. Find the prime factorization for each integer.

2. The GCF is the product of the common prime factors using the smallest exponent that appears on each of them.

If two integers have no common prime factors, then their greatest common factor is 1, because 1 is a factor of every integer. For example, 6 and 35 have no common prime factors because 6 = 2 ⋅ 3 and 35 = 5 ⋅ 7. However, because 6 = 1 ⋅ 6 and 35 = 1 ⋅ 35, the GCF for 6 and 35 is 1.

Find the GCF for each group of numbers.

a)	First find the prime factorization for each number: Because 2 is not a factor of 225, it is not a common factor of 150 and 225. Only 3 and 5 appear in both factorizations. Looking at both 2 ⋅ 3 ⋅ 52 and 32 ⋅ 52, we see that the smallest power of 5 is 2 and the smallest power of 3 is 1. So the GCF for 150 and 225 is 3 ⋅ 52, or 75.
b)	First find the prime factorization for each number: The only common prime factors are 2 and 3. The smallest power of 2 in the factorizations is 3, and the smallest power of 3 is 2. So the GCF is 23 ⋅ 32, or 72.
c)	First find the prime factorization for each number: Because there are no common factors other than 1, the GCF is 1.

Now do Exercises 13–22

To find the GCF for a group of monomials, we use the same procedure as that used for integers.

1. Find the GCF for the coefficients of the monomials.

2. Form the product of the GCF for the coefficients and each variable that is common to all of the monomials, where the exponent on each variable is the smallest power of that variable in any of the monomials.

Prime Factorization of Integers

Find the greatest common factor for each group of monomials.

a)	Since 15 = 3 ⋅ 5 and 9 = 32, the GCF for 15 and 9 is 3. Since the smallest power of x in 15x2 and 9x3 is 2, the GCF is 3x2. If we write these monomials as we can see that 3x2 is the GCF.
b)	Since 12 = 22 ⋅ 3, 30 = 2 ⋅ 3 ⋅ 5, and 42 = 2 ⋅ 3 ⋅ 7, the GCF for 12, 30, and 42 is 2 ⋅ 3 or 6. For the common variables x and y, 2 is the smallest power of x and 1 is the smallest power of y. So the GCF for the three monomials is 6x2y. Note that z is not in the GCF because it is not in all three monomials.

Now do Exercises 23–34

In Chapter 4, we used the distributive property to multiply monomials and polynomials. For example,

If we start with 30x − 18 and write

we have factored 30x − 18. Because multiplication is the last operation to be performed in 6(5x − 3), the expression 6(5x − 3) is a product. Because 6 is the GCF for 30 and 18, we have factored out the GCF.

Factor the following polynomials by factoring out the GCF.

a)	The GCF for the coefficients 25 and 40 is 5. Because the smallest power of the common factor a is 1, we can factor 5a out of each term:
b)	The GCF for 6, 12, and 3 is 3. We can factor x2 out of each term, since the smallest power of x in the three terms is 2. So factor 3x2 out of each term as follows: Check by multiplying: 3x2(2x2 − 4x + 1) = 6x4 − 12x3 + 3x2.
c)	The GCF for the numerical coefficients is 1. Both x and y are common to each term. Using the lowest powers of x and y, we get Check by multiplying.

Now do Exercises 35-62

Because of the commutative property of multiplication, the common factor can be placed on either side of the other factor. So in Example 5, the answers could be written as (5a + 8)5a, (2x2 − 4x + 1)3x2, and (y2 + x4)x2y3.

If the GCF is one of the terms of the polynomial, then you must remember to leave a 1 in place of that term when the GCF is factored out. For example,

You should always check your answer by multiplying the factors.

In Example 6, the greatest common factor is a binomial. This type of factoring will be used in factoring trinomials by grouping in Section 5.2.

Factor out the greatest common factor.

a)	The greatest common factor is a + b:
b)	The greatest common factor is x + 2:
c)	The greatest common factor is y − 3:

Now do Exercises 63–70

The greatest common factor for −4x + 2xy is 2x. Note that you can factor out the GCF (2x) or the opposite of the GCF (−2x):

It is useful to know both of these factorizations. Factoring out the opposite of the GCF will be used in factoring by grouping in Section 5.2 and in factoring trinomials with negative leading coefficients in Section 5.4. Remember to check all factoring by multiplying the factors to see if you get the original polynomial.

Factor each polynomial twice. First factor out the greatest common factor, and then factor out the opposite of the GCF.

a)	Note that the signs of the terms in parentheses change when −3 is factored out. Check the answers by multiplying.
b)	We can also write a − b = −1(b − a).
c)

Now do Exercises 71–86

Be sure to change the sign of each term in parentheses when you factor out the opposite of the greatest common factor.

The area of a rectangular garden is x2 + 8x + 15 square feet. If the length is x + 5 feet, then what binomial represents the width?

Note that the area of a rectangle is the product of the length and width. Since x2 + 8x + 15 = (x + 5)(x + 3) and the length is x + 5 feet, the width must be x + 3 feet.

Now do Exercises 87–90

1. To means to write as a product.

2. A number is an integer greater than 1 that has no factors besides itself and 1.

3. The of two numbers is the largest number that is a factor of both.

4. All factoring can be checked by the factors.

1. There are only nine prime numbers.

2. The prime factorization of 32 is 23 ⋅ 3.

3. The integer 51 is a prime number.

4. The GCF for 12 and 16 is 4.

5. The GCF for x5y3 − x4y7 is x4y3.

6. We can factor out 2xy or −2 xy from 2x2y − 6xy2.

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Find the prime factorization of each integer. See Examples 1 and 2.

1. 18

2. 20

3. 52

4. 76

5. 98

6. 100

7. 216

8. 248

9. 460

10. 345

11. 924

12. 585

Find the greatest common factor for each group of integers. See Example 3. See the Strategy for Finding the GCF for Positive Integers box on page 323.

1. 8, 20

2. 18, 42

3. 36, 60

4. 42, 70

5. 40, 48, 88

6. 15, 35, 45

7. 76, 84, 100

8. 66, 72, 120

9. 39, 68, 77

10. 81, 200, 539

Find the greatest common factor for each group of monomials. See Example 4. See the Strategy for Finding the GCF for Monomials box on page 324.

1. 6x, 8x3

2. 12x2, 4x3

3. 12x3,4x2, 6x

4. 3y5, 9y4, 15y3

5. 3x2y, 2xy2

6. 7a2x3, 5a3x

7. 24a2bc, 60ab2

8. 30x2yz3, 75x3yz6

9. 12u3v2, 25s2t4

10. 45m2n5, 56a4b8

11. 18a3b, 30a2b2, 54ab3

12. 16x2z, 40xz2, 72z3

Complete the factoring of each monomial.

1. 27x = 9

2. 51y = 3y

3. 24t2 = 8t

   Exercise 38 - Complete the Factoring of Each Monomial

4. 18u2 = 3u

5. 36y5 = 4y2

6. 42z4 = 3z2

7. u4v3 = uv

8. x5y3 = x2y

9. −14m4n3 = 2m4

10. −8y3z4 = 4z3

11. −33x4y3z2 = −3x3yz

12. −96a3b4c5 = −12ab3c3

Factor out the GCF in each expression. See Example 5.

1. 2w + 4t

2. 6y + 3

3. 12x − 18y

4. 24a − 36b

5. x3 − 6x

   Exercise 52 - Factor Out GCF of a Binomial

6. 10y4 − 30y2

7. 5ax + 5ay

8. 6wz + 15wa

9. h5 + h3

11. −2k7m4 + 4k3m6

12. −6h5t2 + 3h3t6

13. 2x3 − 6x2 + 8x

14. 6x3 + 18x2 + 24x

15. 12x4t + 30x3t − 24x2t2

16. 15x2y2 − 9xy2 + 6x2y

Factor out the GCF in each expression. See Example 6.

1. (x − 3)a + (x − 3)b

2. (y + 4)3 + (y + 4)z

3. x(x − 1) − 5(x − 1)

   Exercise 66 - GCF is a Binomial

4. a(a + 1) − 3(a + 1)

5. m(m + 9) + (m + 9)

6. (x − 2)x − (x − 2)

7. a(y + 1)2 + b(y + 1)2

8. w(w + 2)2 + 8(w + 2)2

First factor out the GCF, and then factor out the opposite of the GCF. See Example 7.

1. 8x − 8y

2. 2a − 6b

3. −4x + 8x2

4. −5x2 + 10x

5. x − 5

6. a − 6

7. 4 − 7a

8. 7 − 5b

9. −24a3 + 16a2

10. −30b4 + 75b3

11. −12x2 − 18x

12. −20b2 − 8b

13. −2x3 − 6x2 + 14x

    Exercise 84 - Factoring Out the Opposite of the GCF

14. −8x4 + 6x3 − 2x2

15. 4a3b − 6a2b2 − 4ab3

16. 12u5v6 + 18u2v3 − 15u4v5

Solve each problem by factoring. See Example 8.

1. Uniform motion. Helen traveled a distance of 20x + 40 miles at 20 miles per hour on the Yellowhead Highway. Find a binomial that represents the time that she traveled.

2. Area of a painting. A rectangular painting with a width of x centimeters has an area of x2 + 50x square cen timeters. Find a binomial that represents the length. See the accompanying figure.

   Figure for Exercise 88

3. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup depends on the radius r and height h:

   a)	Rewrite this formula by factoring out the greatest common factor on the right-hand side.
   b)	Let h = 5 in. and write a formula that expresses S in terms of r.
   c)	The accompanying graph shows S for r between 1 in. and 3 in. (with h = 5 in.). Which of these r-values gives the maximum surface area?

   Figure for Exercise 89

4. Amount of an investment. The amount of an investment of P dollars for t years at simple interest rate r is given by A = P + Prt.

   a)	Rewrite this formula by factoring out the greatest common factor on the right-hand side.
   b)	Find A if $8300 is invested for 3 years at a simple interest rate of 15%.

1. Discussion

   Is the greatest common factor of −6x2 + 3x positive or negative? Explain.

2. Writing

   Explain in your own words why you use the smallest power of each common prime factor when finding the GCF of two or more integers.

Kayaks have been built by the Aleut and Inuit peoples for the past 4000 years. Today's builders have access to materials and techniques unavailable to the original kayak builders. Modern kayakers incorporate hydrodynamics and materials technology to create designs that are efficient and stable. Builders measure how well their designs work by calculating indicators such as prismatic coefficient, block coefficient, and the midship area coefficient, to name a few.

Even the fitting of a kayak to the paddler is done scientifically. For example, the formula

can be used to calculate the appropriate paddle length. BL is the length of the paddle's blade. BS is a boating style factor, which is 1.2 for touring, 1.0 for river running, and 0.95 for play boating. EE is the elbow to elbow distance with the paddler's arms straight out to the sides. BW is the boat width and SW is the shoulder width. SL is the spine length, which is the distance measured in a sitting position from the chair seat to the top of the paddler's shoulder. All lengths are in centimeters.

The degree of control a kayaker exerts over the kayak depends largely on the body contact with it. A kayaker wears the kayak. So the choice of a kayak should hinge first on the right body fit and comfort and second on the skill level or intended paddling style. So designing, building, and even fitting a kayak is a blend of art and science.