Domains of Rational Expressions

In Section 5.1 you learned how to factor out the greatest common factor from allof the terms of a polynomial. In this section you will learn to factor a four-term polynomial by factoring out a common factor from the first two terms and then a common factor from the last two terms.

The product of two binomials may have four terms. For example,

To factor x2 + ax + 3x + 3a, we simply reverse the steps we used to find the product. Factor out the common factor x from the first two terms and the common factor 3 from the last two terms:

It does not matter whether you take out the common factor to the right or left. So(x + a)(x + 3) is also correct and we could have factored as follows:

This method of factoring is called factoring by grouping.

1. Factor out the GCF from the first group of two terms.

2. Factor out the GCF from t he last group of two terms.

3. Factor out the common binomial.

Factor by Grouping

Use grouping to factor each polynomial.

a)	The first two terms have a common factor of y, and the last two terms have acommon factor of 5: Check by using FOIL.
b)	The first two terms have a common factor of x, and the last two have a common factor of 1: Check by using FOIL.

Now do Exercises 1–10

For some four-term polynomials it is necessary to rearrange the terms before factoring out the common factors.

Use grouping to factor each polynomial.

a)	We can factor out m from the first two terms to get m(n + 4), but we can't get another factor of n + 4 from the last two terms. By rearranging the terms we can factor by grouping:
b)

Now do Exercises 11–18

Note that there are several rearrangements that will allow us to factor the polynomials in Example 2. For example, m2 + 4m + mn + 4n would also work for Example 2(a).

We saw in Section 5.1 that you could factor out a common factor with a positive sign or a negative sign. For example, we can factor −2x + 10 as 2(−x + 5) or −2(x − 5). We use this technique in Example 3.

Use grouping to factor each polynomial.

a)	We can factor out x from the first two terms and 1 from the last two terms: However, we didn't get a common binomial. We can get a common binomial if we factor out −1 from the last two terms:
b)	For this polynomial we have to rearrange the terms and factor out a common factor with a negative sign:

Now do Exercises 19–28

In Section 4.7, you learned that the product of a sum and a difference is a difference of two squares:

So a difference of two squares can be factored as a product of a sum and a difference, using the following rule.

For any real numbers a and b,

Note that the square of an integer is a perfect square. For example, 64 is a perfect square because 64 = 82. The square of a monomial in which the coefficient is an integer is also called a perfect square or simply a square. For example, 9m2 is a perfect square because 9m2 = (3m)2.

Factoring a Difference of Two Squares

Factor each polynomial.

a)	Because 81 = 92, the binomial y2 − 81 is a difference of two squares: Check by multiplying.
b)	Because 9m2 = (3m)2 and 16 = 42, the binomial 9m2 − 16 is a difference of two squares: Check by multiplying.
c)	Because 4x2 = (2x)2 and 9y2 = (3y)2, the binomial 4x2 − 9y2 is a difference of two squares:

Now do Exercises 29–42

Don't confuse a difference of two squares a2 − b2 with a sum of two squares a2 + b2. The sum a2 + b2 is not one of the special products and it can't be factored.

In Section 4.7 you learned how to square a binomial using the rule

You can reverse this rule to factor a trinomial such as x2 + 6x + 9. Notice that

So if a = x and b = 3, then x2 + 6x + 9 fits the form a2 + 2ab + b2, and

A trinomial that is of the form a2 + 2ab + b2 or a2−2ab + b2 is called a perfect square trinomial. A perfect square trinomial is the square of a binomial. Perfect square trinomials will be used in solving quadratic equations by completing the square in Chapter 10. Perfect square trinomials can be identified using the following strategy.

A trinomial is a perfect square trinomial if

1. the first and last terms are of the form a2 and b2 (perfect squares), and

2. the middle term is 2ab or −2ab.

Determine whether each binomial is a difference of two squares and whether each trinomial is a perfect square trinomial.

Now do Exercises 43–54

Note that the middle term in a perfect square trinomial may have a positive or a negative coefficient, while the first and last terms must be positive. Any perfect square trinomial can be factored as the square of a binomial by using the following rule.

For any real numbers a and b,

Factor.

a)	The first term is x2, and the last term is 22. Because the middle term is −2 ⋅ 2 ⋅ x, or −4x, this polynomial is a perfect square trinomial: Check by expanding (x−2)2.
b)	a2 + 16a + 64 = (a + 8)2 Check by expanding (a + 8)2.
c)	The first term is (2x)2, and the last term is 32. Because −2 ⋅ 2x ⋅ 3 = −12x, the polynomial is a perfect square trinomial. So Check by expanding (2x − 3)2.

Now do Exercises 55–72

To factor a polynomial means to write it as a product of simpler polynomials. A polynomial that can't be factored using integers is called a prime or irreduciblepolynomial. The polynomials 3x, w + 1, and 4m − 5 are prime polynomials. Note that , but 4m − 5 is a prime polynomial because it can't be factored using integers only.

A polynomial is factored completely when it is written as a product of prime polynomials. So (y−8)(y + 1) is a complete factorization. When factoring polynomials, we usually do not factor integers that occur as common factors. So 6x(x−7) is considered to be factored completely even though 6 could be factored.

Some polynomials have a factor common to all terms. To factor such polynomials completely, it is simpler to factor out the greatest common factor (GCF) and thenfactor the remaining polynomial. Example 7 illustrates factoring completely.

Factor each polynomial completely.

a)	The greatest common factor of 2x3 and 50x is 2x:
b)
c)	We can factor out x2 from the first two terms and 1 from the last two terms: However, we didn't get a common binomial. We can get a common binomial if we factor out −1 from the last two terms:

Now do Exercises 73–98

Remember that factoring reverses multiplication and every step of factoring can be checked by multiplication.

1. A is the square of an integer or an algebraic expression.

2. A is the product of a sum and a difference.

3. A trinomial of the form a2 + 2ab + b2 is a trinomial.

4. A polynomial is one that can't be factored.

5. A polynomial is when it is written as a product of prime polynomials.

1. We always factor out the GCF first.

2. The polynomial x2 + 16 is a difference of two squares.

3. The polynomial x2 − 8x + 16 is a perfect square trinomial.

4. The polynomial 9x2 + 21x + 49 is a perfect square trinomial.

5. The polynomial 16y + 1 is a prime polynomial.

6. The polynomial 4x2 − 4 is factored completely as 4(x2 − 1).

• As you study a chapter, make a list of topics and questions that you would put on the test, if you were to write it.

• Write about what you read in the text. Sum things up in your own words.

Factor by grouping. See Example 1.

1. bx + by + cx + cy

2. 3x + 3z + ax + az

3. ab + b2 + a + b

4. 2x2 + x + 2x + 1

5. wm + 3w + m + 3

6. ay + y + 3a + 3

7. 6x2 + 10x + 3xw + 5w

8. 5ax + 2ay + 5xy + 2y2

9. x2 + 3x + 4x + 12

10. y2 + 2y + 6y + 12

Factor by grouping. See Example 2.

1. mn + n + n2 + m

2. 2x3 + y + x + 2x2y

3. 10 + wm + 5m + 2w

4. 2a + 3b + 6 + ab

   Exercise 15 - Factor by Grouping

5. xa + ay + 3y + 3x

6. x3 + ax + 3a + 3x2

• a3 + w2 + aw + a2w

• a4 + y + ay + a3

Factor by grouping. See Example 3.

1. w2 − w − bw + b

2. x2 − 2x − mx + 2m

3. w2 + aw − w − a

4. ap + 3a − p − 3

5. m2 + mx − x − m

6. −6n − 6b + b2 + bn

7. x2 + 7x − 5x − 35

8. y2 + 3y − 8y − 24

9. 2x2 + 14x − 5x − 35

10. 2y2 + 3y − 16y − 24

Factor each polynomial. See Example 4.

1. a2 − 4

2. h2 − 9

3. x2 − 49

   Exercise 32 - Factoring a Difference of Two Squares

4. y2 − 36

5. a2 − 121

6. w2 − 81

7. y2 − 9x2

8. 16x2 − y2

9. 25a2 − 49b2

10. 9a2 − 64b2

11. 121m2 − 1

12. 144n2 − 1

13. 9w2 − 25c2

14. 144w2 − 121a2

Determine whether each polynomial is a difference of two squares, a perfect square trinomial, or neither of these. See Example 5. See the Strategy for Identifying Perfect Square Trinomials box on page 334.

1. x2 − 20x + 100

2. x2 − 10x − 25

3. y2 − 40

4. a2 − 49

5. 4y2 + 12y + 9

6. 9a2 − 30a − 25

7. x2 − 8x + 64

8. x2 + 4x + 4

9. 9y2 − 25c2

10. 9x2 + 4

11. 9a2 + 6ab + b2

12. 4x2 − 4xy + y2

Factor each perfect square trinomial. See Example 6.

9. Exercise 64 - Factoring a Perfect Square Trinomial

Factor each polynomial completely. See Example 7.

5. Exercise 78 - Factoring Completely: GCF is a variable

13. −3x2 + 3y2)

14. −8a2 + 8b2

    Exercise 87 - Factoring Completely

15. 2ax2 − 98a

16. 32x2y - 2y3)

17. w3 − w − w2 + 1

18. x3 + x2 − x − 1

19. x3 + x2 − 4x − 4

20. a2m − b2n + a2n − b2m

21. 3ab2 − 18ab + 27a

22. −2a2b + 8ab − 8b

23. −4m3 + 24m2n − 36mn2

24. 10a3 − 20a2b + 10ab2

25. x2a − b + bx2 − a

26. wx2 − 75 − 25w + 3x2

Factor each polynomial completely.

1. 6a3y + 24a2y2 + 24ay3

2. 8b5c − 8b4c2 + 2b3c3

3. 24a3y − 6ay3

4. 27b3c − 12bc3

5. 2a3y2 − 6a2y

6. 9x3y − 18x2y2

7. ab + 2bw − 4aw − 8w2

8. 3am − 6n − an + 18m

9. (a − b) − b(a − b)

10. (a + b)w − (a + b)

11. (4x2 − 1)2x − (4x2 − 1)

12. (a2 − 9)a + 3(a2 − 9)

Use factoring to solve each problem.

1. Skydiving. The height in feet above the earth for askydiver t seconds after jumping from an airplane at 6400 ft is approximated by the formula h(t) = − 16t2 + 6400, provided t < 5.

   a)	Rewrite the formula with the right-hand side factored completely.
   b)	Use the result of part (a) to find h(2).

   Figure for Exercise 111

2. Demand for pools. Tropical Pools sells an aboveground model for p dollars each. The monthly revenue for this model is given by the formula

   Revenue is the product of the price p and the demand (quantity sold).

   a)	Factor out the price on the right-hand side of the formula. R(p) = p(−0.08p + 300)
   b)	Write a formula D(p) for the monthly demand.
   c)	Find D(3000).
   d)	Use the accompanying graph to estimate the price at which the revenue is maximized. Approximately how many pools will be sold monthly at this price?
   e)	What is the approximate maximum revenue?
   f)	Use the accompanying graph to estimate the price at which the revenue is zero.

   Figure for Exercise 112

3. Volume of a tank. The volume in cubic inches for a fish tank with a square base and height x is given by the formula

   a)	Rewrite the formula with the right-hand side factored completely.
   b)	Find an expression for the length of a side of the square base.

   Figure for Exercise 113

1. Discussion

   For what real number k does 3x2 − k factor as 3(x − 2)(x + 2)?

2. Writing

   Explain in your own words how to factor a four-term polynomial by grouping.

3. Writing

   Explain how you know that x2 + 1 is a prime polynomial.