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24 Chapter P Prerequisites

P.3 Polynomials and Factoring

Polynomials What you should learn ᭿ Write polynomials in standard form. An algebraic expression is a collection of variables and real numbers. The most ᭿ Add,subtract,and multiply polynomials. polynomial. common type of algebraic expression is the Some examples are ᭿ Use special products to multiply polyno- 2x ϩ 5, 3x 4 Ϫ 7x 2 ϩ 2x ϩ 4, and 5x 2y 2 Ϫ xy ϩ 3. mials. ᭿ Remove common factors from polynomi- The first two are polynomials in x and the third is a polynomial in x and y. The als. terms of a polynomial in x have the form ax k, where a is the coefficient and k is ᭿ Factor special polynomial forms. the degree of the term. For instance, the polynomial ᭿ Factor trinomials as the product of two binomials. 3 Ϫ 2 ϩ ϭ 3 ϩ ͑Ϫ ͒ 2 ϩ ͑ ͒ ϩ 2x 5x 1 2x 5 x 0 x 1 ᭿ Factor by grouping. has coefficients 2,Ϫ5, 0, and 1. Why you should learn it Polynomials can be used to model and solve Definition of a Polynomial in x real-life problems.For instance,in Exercise 157 on page 34,a polynomial is used to Let a0, a1, a2, . . . , an be real numbers and let n be a nonnegative integer. model the total distance an automobile A polynomial in x isP.3 an expression of the form travels when stopping. n ϩ nϪ1 ϩ . . . ϩ ϩ an x anϪ1x a1x a 0  where an 0. The polynomial is of degree n, an is the leading coeffi- cient, and a0 is the constant term.

In standard form, a polynomial in x is written with descending powers of x. Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polyno- mials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree of its terms. For instance, the degree of the polynomial Ϫ2x3y6 ϩ 4xy Ϫ x7y4 is 11 because the sum of the exponents in the last term is the greatest. Expressions © Robert W. Ginn/age fotostock such as the following are not polynomials.

͞ x3 Ϫ Ί3x ϭ x3 Ϫ ͑3x͒1 2 The exponent 1͞2 is not an integer.

5 Ϫ x2 ϩ ϭ x2 ϩ 5x 1 The exponent Ϫ1 is not a nonnegative integer. STUDY TIP x Expressions are not polynomials Example 1 Writing Polynomials in Standard Form if: Polynomial Standard Form Degree 1. A variable is underneath a radical. a. 4x 2 Ϫ 5x 7 Ϫ 2 ϩ 3x Ϫ5x 7 ϩ 4x 2 ϩ 3x Ϫ 2 7 2. A polynomial expression b. 4 Ϫ 9x 2 Ϫ9x 2 ϩ 4 2 (with degree greater than 0) c. 88͑8 ϭ 8x 0͒ 0 is in the denominator of a Now try Exercise 15. term. 333371_0P03.qxp 12/27/06 9:31 AM Page 25

Section P.3 Polynomials and Factoring 25 Operations with Polynomials You can add and subtract polynomials in much the same way you add and sub- tract real numbers. Simply add or subtract the like terms (terms having exactly the same variables to exactly the same powers) by adding their coefficients. For instance,Ϫ3xy 2 and 5xy2 are like terms and their sum is Ϫ3xy2 ϩ 5xy2 ϭ ͑Ϫ3 ϩ 5͒xy2 ϭ 2xy2.

Example 2 Sums and Differences of Polynomials STUDY TIP Perform the indicated operation. When a negative sign precedes a. ͑5x 3 Ϫ 7x 2 Ϫ 3͒ ϩ ͑x 3 ϩ 2x 2 Ϫ x ϩ 8͒ an expression within parentheses, ͑ 4 Ϫ 2 Ϫ ϩ ͒ Ϫ ͑ 4 Ϫ 2 ϩ ͒ b. 7x x 4x 2 3x 4x 3x treat it like the coefficient ͑Ϫ1͒ and distribute the negative sign Solution to each term inside the a. ͑5x 3 Ϫ 7x 2 Ϫ 3͒ ϩ ͑x 3 ϩ 2x 2 Ϫ x ϩ 8͒ parentheses. ϭ ͑5x 3 ϩ x 3͒ ϩ ͑Ϫ7x 2 ϩ 2x 2͒ Ϫ x ϩ ͑Ϫ3 ϩ 8͒ Group like terms. Ϫ͑x2 Ϫ x ϩ 3͒ ϭϪx2 ϩ x Ϫ 3 ϭ 6x 3 Ϫ 5x 2 Ϫ x ϩ 5 Combine like terms. b. ͑7x 4 Ϫ x2 Ϫ 4x ϩ 2͒ Ϫ ͑3x 4 Ϫ 4x2 ϩ 3x͒ ϭ 7x 4 Ϫ x2 Ϫ 4x ϩ 2 Ϫ 3x 4 ϩ 4x2 Ϫ 3x Distributive Property ϭ ͑7x 4 Ϫ 3x 4͒ ϩ ͑Ϫx2 ϩ 4x2͒ ϩ ͑Ϫ4x Ϫ 3x͒ ϩ 2 Group like terms. ϭ 4x 4 ϩ 3x2 Ϫ 7x ϩ 2 Combine like terms. Now try Exercise 23.

To find the product of two polynomials, use the left and right Distributive Properties.

Example 3 Multiplying Polynomials: The FOIL Method When using the FOIL method, the following scheme may be helpful. ͑3x Ϫ 2͒͑5x ϩ 7͒ ϭ 3x͑5x ϩ 7͒ Ϫ 2͑5x ϩ 7͒ F ϭ ͑3x͒͑5x͒ ϩ ͑3x͒͑7͒ Ϫ ͑2͒͑5x͒ Ϫ ͑2͒͑7͒ L 2 ϭ 15x ϩ 21x Ϫ 10x Ϫ 14 ͑3x Ϫ 2͒͑5x ϩ 7͒

I Product of Product of Product of Product of O First terms Outer terms Inner terms Last terms

ϭ 15x 2 ϩ 11x Ϫ 14 Note that when using the FOIL Method (which can be used only to multiply two binomials), the outer (O) and inner (I) terms may be like terms that can be combined into one term. Now try Exercise 39. 333371_0P03.qxp 12/27/06 9:31 AM Page 26

26 Chapter P Prerequisites

Example 4 The Product of Two Trinomials Find the product of 4x2 ϩ x Ϫ 2 and Ϫx2 ϩ 3x ϩ 5. Solution When multiplying two polynomials, be sure to multiply each term of one polynomial by each term of the other. A vertical format is helpful.

4 x2 ϩ x Ϫ 2 Write in standard form.

ϫ Ϫx2 ϩ 3x ϩ 5 Write in standard form.

2 0 x2 ϩ 5x Ϫ 10 5͑4x2 ϩ x Ϫ 2͒

1 2 x3 ϩ 3x2 Ϫ 6x 3x͑4x2 ϩ x Ϫ 2͒

Ϫ4x4 Ϫ x3 ϩ 2x2 Ϫx2͑4x2 ϩ x Ϫ 2͒

Ϫ4x4 ϩ 11x3 ϩ 25x2 Ϫ x Ϫ 10 Combine like terms. Now try Exercise 59.

Special Products

Special Products Let u and v be real numbers, variables, or algebraic expressions. Special Product Example Sum and Difference of Same Terms ͑u ϩ v͒͑u Ϫ v͒ ϭ u 2 Ϫ v 2 ͑x ϩ 4͒͑x Ϫ 4͒ ϭ x 2 Ϫ 42 ϭ x2 Ϫ 16 Square of a Binomial ͑u ϩ v͒2 ϭ u 2 ϩ 2uv ϩ v 2 ͑x ϩ 3͒2 ϭ x 2 ϩ 2͑x͒͑3͒ ϩ 32 ϭ x2 ϩ 6x ϩ 9 ͑u Ϫ v͒2 ϭ u 2 Ϫ 2uv ϩ v 2 ͑3x Ϫ 2͒2 ϭ ͑3x͒2 Ϫ 2͑3x͒͑2͒ ϩ 22 ϭ 9x2 Ϫ 12x ϩ 4 Cube of a Binomial ͑u ϩ v͒3 ϭ u 3 ϩ 3u 2v ϩ 3uv 2 ϩ v 3 ͑x ϩ 2͒3 ϭ x 3 ϩ 3x 2͑2͒ ϩ 3x͑22͒ ϩ 23 ϭ x3 ϩ 6x2 ϩ 12x ϩ 8 ͑u Ϫ v͒3 ϭ u 3 Ϫ 3u 2v ϩ 3uv 2 Ϫ v 3 ͑x Ϫ 1͒3 ϭ x 3 Ϫ 3x 2͑1͒ ϩ 3x͑12͒ Ϫ 13 ϭ x3 Ϫ 3x2 ϩ 3x Ϫ 1

Example 5 The Product of Two Trinomials To understand the individual patterns of special products, have students “derive” Find the product of x ϩ y Ϫ 2 and x ϩ y ϩ 2. each product. Then explain how special products save time and how the pattern of Solution the product must be recognized to factor By grouping x ϩ y in parentheses, you can write the product of the trinomials an expression. as a special product. ͑x ϩ y Ϫ 2͒͑x ϩ y ϩ 2͒ ϭ ͓͑x ϩ y͒ Ϫ 2͔͓͑x ϩ y͒ ϩ 2͔ ϭ ͑x ϩ y͒ 2 Ϫ 22 ϭ x 2 ϩ 2xy ϩ y 2 Ϫ 4 Now try Exercise 61. 333371_0P03.qxp 12/27/06 9:31 AM Page 27

Section P.3 Polynomials and Factoring 27 Factoring The process of writing a polynomial as a product is called factoring. It is an important tool for solving equations and for simplifying rational expressions. Unless noted otherwise, when you are asked to factor a polynomial, you can assume that you are looking for factors with integer coefficients. If a polynomial cannot be factored using integer coefficients, it is prime or irreducible over the integers. For instance, the polynomial x 2 Ϫ 3 is irreducible over the integers. Over the real numbers, this polynomial can be factored as x 2 Ϫ 3 ϭ ͑x ϩ Ί3͒͑x Ϫ Ί3͒. A polynomial is completely factored when each of its factors is prime. So, x 3 Ϫ x 2 ϩ 4x Ϫ 4 ϭ ͑x Ϫ 1͒͑x 2 ϩ 4͒ Completely factored Activities is completely factored, but 1. Explain what happens when the paren- theses are removed from the expression: 3 Ϫ 2 Ϫ ϩ ϭ ͑ Ϫ ͒͑ 2 Ϫ ͒ Not completely factored x x 4x 4 x 1 x 4 Ϫ͑3x2 Ϫ 4x ϩ 2͒. is not completely factored. Its complete factorization is Answer: The sign of each term changes. Ϫ3x2 ϩ 4x Ϫ 2 x 3 Ϫ x 2 Ϫ 4x ϩ 4 ϭ ͑x Ϫ 1͒͑x ϩ 2͒͑x Ϫ 2͒. 2. Multiply using special products: 2 The simplest type of factoring involves a polynomial that can be written as ͑Ί2 ϩ Ί3 ͒ . the product of a monomial and another polynomial. The technique used here is Answer: 5 ϩ 2Ί6 ͑ ϩ ͒ ϭ ϩ the Distributive Property,a b c ab ac, in the reverse direction. For 3. Multiply: ͑2x ϩ 1͒͑x Ϫ 5͒. 2 ϩ instance, the polynomial 5x 15x can be factored as follows. Answer: 2x2 Ϫ 9x Ϫ 5 5 x2 ϩ 15x ϭ 5x͑x͒ ϩ 5x͑3͒ 5x is a common factor. ϭ 5x͑x ϩ 3͒ The first step in completely factoring a polynomial is to remove (factor out) any common factors, as shown in the next example.

Example 6 Removing Common Factors Factor each expression. a.6x3 Ϫ 4x b.3x 4 ϩ 9x3 ϩ 6x2 c. ͑x Ϫ 2͒͑2x͒ ϩ ͑x Ϫ 2͒͑3͒

Solution a. 6x3 Ϫ 4x ϭ 2x͑3x2͒ Ϫ 2x͑2͒ ϭ 2x͑3x2 Ϫ 2͒ 2x is a common factor.

b. 3 x 4 ϩ 9x3 ϩ 6x2 ϭ 3x 2͑x2͒ ϩ 3x 2͑3x͒ ϩ 3x2͑2͒ 3x2 is a common factor. ϭ 3x 2͑x2 ϩ 3x ϩ 2͒ ϭ 3x2͑x ϩ 1͒͑x ϩ 2͒

c. ͑x Ϫ 2͒͑2x͒ ϩ ͑x Ϫ 2͒͑3͒ ϭ ͑x Ϫ 2͒͑2x ϩ 3͒ x Ϫ 2 is a common factor. Now try Exercise 73.

Factoring Special Polynomial Forms Some polynomials have special forms that arise from the special product forms on page 26. You should learn to recognize these forms so that you can factor such polynomials easily. 333371_0P03.qxp 12/27/06 9:31 AM Page 28

28 Chapter P Prerequisites

Factoring Special Polynomial Forms Factored Form Example Difference of Two Squares u 2 Ϫ v 2 ϭ ͑u ϩ v͒͑u Ϫ v͒ 9x2 Ϫ 4 ϭ ͑3x͒2 Ϫ 22 ϭ ͑3x ϩ 2͒͑3x Ϫ 2͒ Perfect Square Trinomial u 2 ϩ 2uv ϩ v 2 ϭ ͑u ϩ v͒2 x2 ϩ 6x ϩ 9 ϭ x2 ϩ 2͑x͒͑3͒ ϩ 32 ϭ ͑x ϩ 3͒2 u 2 Ϫ 2uv ϩ v 2 ϭ ͑u Ϫ v͒2 x 2 Ϫ 6x ϩ 9 ϭ x 2 Ϫ 2͑x͒͑3͒ ϩ 32 ϭ ͑x Ϫ 3͒2 Sum or Difference of Two Cubes u 3 ϩ v 3 ϭ ͑u ϩ v͒͑u 2 Ϫ uv ϩ v 2͒ x 3 ϩ 8 ϭ x 3 ϩ 23 ϭ ͑x ϩ 2͒͑x2 Ϫ 2x ϩ 4͒ u 3 Ϫ v 3 ϭ ͑u Ϫ v͒͑u 2 ϩ uv ϩ v 2͒ 27x3 Ϫ 1 ϭ ͑3x͒3 Ϫ 13 ϭ ͑3x Ϫ 1͒͑9x2 ϩ 3x ϩ 1͒

One of the easiest special polynomial forms to factor is the difference of two squares. Think of this form as follows. u 2 Ϫ v 2 ϭ ͑u ϩ v͒͑u Ϫ v͒

Difference Opposite signs To recognize perfect square terms, look for coefficients that are squares of integers and variables raised to even powers.

Example 7 Removing a Common Factor First STUDY TIP 3 Ϫ 12x 2 ϭ 3͑1 Ϫ 4x2͒ 3 is a common factor.

ϭ 3͓12 Ϫ ͑2x͒2͔ Difference of two squares In Example 7, note that the first step in factoring a polynomial is ϭ ͑ ϩ ͒͑ Ϫ ͒ 3 1 2x 1 2x Factored form to check for a common factor. Now try Exercise 77. Once the common factor is removed, it is often possible to recognize patterns that were not Example 8 Factoring the Difference of Two Squares immediately obvious.

a. ͑x ϩ 2͒2 Ϫ y2 ϭ ͓͑x ϩ 2͒ ϩ y͔͓͑x ϩ 2͒ Ϫ y͔ ϭ ͑x ϩ 2 ϩ y͒͑x ϩ 2 Ϫ y͒

b. 16x 4 Ϫ 81 ϭ ͑4x2͒2 Ϫ 92 Difference of two squares ϭ ͑4x2 ϩ 9͒͑4x2 Ϫ 9͒

ϭ ͑4x2 ϩ 9͓͒͑2x͒2 Ϫ 32͔ Difference of two squares

ϭ ͑4x2 ϩ 9͒͑2x ϩ 3͒͑2x Ϫ 3͒ Factored form Now try Exercise 81. 333371_0P03.qxp 12/27/06 9:31 AM Page 29

Section P.3 Polynomials and Factoring 29

A perfect square trinomial is the square of a binomial, as shown below. u2 ϩ 2uv ϩ v2 ϭ ͑u ϩ v͒2 or u2 Ϫ 2uv ϩ v2 ϭ ͑u Ϫ v͒2

Like signs Like signs Note that the first and last terms are squares and the middle term is twice the product of u and v.

Example 9 Factoring Perfect Square Trinomials Factor each trinomial. a.x2 Ϫ 10x ϩ 25 b. 16x2 ϩ 8x ϩ 1

Solution a. x 2 Ϫ 10x ϩ 25 ϭ x 2 Ϫ 2͑x͒͑5͒ ϩ 5 2 Rewrite in u2 Ϫ 2uv ϩ v2 form. ϭ ͑x Ϫ 5͒2

b. 16 x 2 ϩ 8x ϩ 1 ϭ ͑4x͒ 2 ϩ 2͑4x͒͑1͒ ϩ 12 Rewrite in u2 ϩ 2uv ϩ v2 form. ϭ ͑4x ϩ 1͒2 Now try Exercise 87.

The next two formulas show the sums and differences of cubes. Pay special attention to the signs of the terms.

Like signs Like signs Exploration u 3 ϩ v 3 ϭ ͑u ϩ v͒͑u 2 Ϫ uv ϩ v 2͒ u 3 Ϫ v 3 ϭ ͑u Ϫ v͒͑u 2 ϩ uv ϩ v 2͒ Rewrite u6 Ϫ v6 as the differ- ence of two squares. Then find a formula for completely factor- Unlike signs Unlike signs ing u 6 Ϫ v 6. Use your formula to factor completely x 6 Ϫ 1 and Example 10 Factoring the Difference of Cubes x6 Ϫ 64. Factor x3 Ϫ 27. Solution x 3 Ϫ 27 ϭ x3 Ϫ 33 Rewrite 27 as 33.

ϭ ͑x Ϫ 3͒͑x 2 ϩ 3x ϩ 9͒ Factor. Now try Exercise 92.

Example 11 Factoring the Sum of Cubes 3x3 ϩ 192 ϭ 3͑x3 ϩ 64͒ 3 is a common factor. ϭ 3͑x 3 ϩ 43͒ Rewrite 64 as 43.

ϭ 3͑x ϩ 4͒͑x 2 Ϫ 4x ϩ 16͒ Factor. Now try Exercise 93. 333371_0P03.qxp 12/27/06 9:31 AM Page 30

30 Chapter P Prerequisites

Trinomials with Binomial Factors You can draw arrows to find the correct middle term. (Encourage students to find To factor a trinomial of the form ax 2 ϩ bx ϩ c, use the following pattern. the middle term mentally.) ͑ x Ϫ ͒͑x ϩ ͒ Factors of a 2 7 15

Ϫ7x ax2 ϩ bx ϩ c ϭ ͑᭿x ϩ ᭿͒͑᭿x ϩ ᭿͒ ϩ30x

Factors of c ϩ23x Middle term

The goal is to find a combination of factors of a and c such that the outer and ͑2x Ϫ 3͒͑x ϩ 5͒ inner products add up to the middle term bx. For instance, in the trinomial 6x 2 ϩ 17x ϩ 5, you can write all possible factorizations and determine which Ϫ one has outer and inner products that add up to 17x. 3x

͑6x ϩ 5͒͑x ϩ 1͒, ͑6x ϩ 1͒͑x ϩ 5͒, ͑2x ϩ 1͒͑3x ϩ 5͒, ͑2x ϩ 5͒͑3x ϩ 1͒ ϩ10x You can see that ͑2x ϩ 5͒͑3x ϩ 1͒ is the correct factorization because the outer ϩ7x Middle term (O) and inner (I) products add up to 17x.

FOIL O ϩ I ͑2x ϩ 5͒͑3x ϩ 1͒ ϭ 6x 2 ϩ 2x ϩ 15x ϩ 5 ϭ 6x 2 ϩ 17x ϩ 5. Point out to students that “testing the middle term” means using the FOIL method to find the sum of the outer and inner terms. Example 12 Factoring a Trinomial: Leading Coefficient Is 1 Factor x 2 Ϫ 7x ϩ 12. STUDY TIP Solution The possible factorizations are Factoring a trinomial can involve trial and error. However, ͑x Ϫ 2͒͑x Ϫ 6͒, ͑x Ϫ 1͒͑x Ϫ 12͒, and ͑x Ϫ 3͒͑x Ϫ 4͒. once you have produced the Testing the middle term, you will find the correct factorization to be factored form, it is an easy matter to check your answer. x 2 Ϫ 7x ϩ 12 ϭ ͑x Ϫ 3͒͑x Ϫ 4͒. O ϩ I ϭϪ4x ϩ ͑Ϫ3x͒ ϭϪ7x For instance, you can verify the Now try Exercise 103. factorization in Example 12 by multiplying out the expression ͑x Ϫ 3͒͑x Ϫ 4͒ to see that you Example 13 Factoring a Trinomial: Leading Coefficient obtain the original trinomial, Is Not 1 x 2 Ϫ 7x ϩ 12. Factor 2x 2 ϩ x Ϫ 15. Solution The eight possible factorizations are as follows. ͑2x Ϫ 1͒͑x ϩ 15͒, ͑2x ϩ 1͒͑x Ϫ 15͒, ͑2x Ϫ 3͒͑x ϩ 5͒, ͑2x ϩ 3͒͑x Ϫ 5͒,

͑2x Ϫ 5͒͑x ϩ 3͒, ͑2x ϩ 5͒͑x Ϫ 3͒, ͑2x Ϫ 15͒͑x ϩ 1͒, ͑2x ϩ 15͒͑x Ϫ 1͒ Testing the middle term, you will find the correct factorization to be

2x 2 ϩ x Ϫ 15 ϭ ͑2x Ϫ 5͒͑x ϩ 3͒. O ϩ I ϭ 6x Ϫ 5x ϭ x Now try Exercise 111. 333371_0P03.qxp 12/27/06 9:32 AM Page 31

Section P.3 Polynomials and Factoring 31 Factoring by Grouping Sometimes polynomials with more than three terms can be factored by a method called factoring by grouping.

Example 14 Factoring by Grouping STUDY TIP Use factoring by grouping to factor x3 Ϫ 2x2 Ϫ 3x ϩ 6. When grouping terms be sure to Solution strategically group terms that x3 Ϫ 2x2 Ϫ 3x ϩ 6 ϭ ͑x3 Ϫ 2x2͒ Ϫ ͑3x Ϫ 6͒ Group terms. have a common factor.

ϭ x 2͑x Ϫ 2͒ Ϫ 3͑x Ϫ 2͒ Factor groups.

ϭ ͑x Ϫ 2͒͑x2 Ϫ 3͒ ͑x Ϫ 2͒ is a common factor. Now try Exercise 115.

Factoring a trinomial can involve quite a bit of trial and error. Some of this trial and error can be lessened by using factoring by grouping. The key to this method of factoring is knowing how to rewrite the middle term. In general, to factor a trinomial ax2 ϩ bx ϩ c by grouping, choose factors of the product ac that add up to b and use these factors to rewrite the middle term.

Example 15 Factoring a Trinomial by Grouping Use factoring by grouping to factor 2x2 ϩ 5x Ϫ 3. Solution In the trinomial 2x 2 ϩ 5x Ϫ 3, a ϭ 2 and c ϭϪ3, which implies that the product ac is Ϫ6. Now, because Ϫ6 factors as ͑6͒͑Ϫ1͒ and 6 Ϫ 1 ϭ 5 ϭ b, rewrite the middle term as 5x ϭ 6x Ϫ x. This produces the following.

2x2 ϩ 5x Ϫ 3 ϭ 2x2 ϩ 6x Ϫ x Ϫ 3 Rewrite middle term.

ϭ ͑2x2 ϩ 6x͒ Ϫ ͑x ϩ 3͒ Group terms. Activities ϭ 2x͑x ϩ 3͒ Ϫ ͑x ϩ 3͒ Factor groups. 1. Completely factor 4x2 Ϫ 4x ϩ 1. Answer: ͑2x Ϫ 1͒2 ϭ ͑x ϩ 3͒͑2x Ϫ 1͒ ͑x ϩ 3͒ is a common factor. 2. Completely factor 9x 4 Ϫ 144. So, the trinomial factors as 2x2 ϩ 5x Ϫ 3 ϭ ͑x ϩ 3͒͑2x Ϫ 1͒. Answer: 9͑x2 ϩ 4͒͑x Ϫ 2͒͑x ϩ 2͒ Now try Exercise 117.

Guidelines for Factoring Polynomials 1. Factor out any common factors using the Distributive Property. 2. Factor according to one of the special polynomial forms. 3. Factor as ax2 ϩ bx ϩ c ϭ ͑mx ϩ r͒͑nx ϩ s͒. 4. Factor by grouping. 333371_0P03.qxp 12/27/06 9:32 AM Page 32

32 Chapter P Prerequisites

P.3 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

Vocabulary Check Fill in the blanks.

n ϩ nϪ1 ϩ . . . ϩ ϩ 1. For the polynomial anx anϪ1x a1x a0, the degree is ______and the leading coefficient is ______. 2. A polynomial that has all zero coefficients is called the ______. 3. A polynomial with one term is called a ______. 4. The letters in “FOIL” stand for the following. F ______O ______I ______L ______5. If a polynomial cannot be factored using integer coefficients, it is called ______. 6. The polynomial u2 ϩ 2uv ϩ v2 is called a ______.

In Exercises 1–6, match the polynomial with its description. In Exercises 21–36, perform the operations and write the [The polynomials are labeled (a), (b), (c), (d), (e), and (f).] result in standard form. a)6x (b) 1 ؊ 4x3 21. ͑6x ϩ 5͒ Ϫ ͑8x ϩ 15͒) c)x3 1 2x2 ؊ 4x 1 1 (d) 7 22. ͑2x 2 ϩ 1͒ Ϫ ͑x 2 Ϫ 2x ϩ 1͒) ؊ 5 3 3 4 2 23. Ϫ͑t3 Ϫ 1͒ ϩ ͑6t3 Ϫ 5t͒ (e)3x 1 2x 1 x (f) 4 x 1 x 1 14 1. A polynomial of degree zero 24. Ϫ͑5x 2 Ϫ 1͒ Ϫ ͑Ϫ3x 2 ϩ 5͒ 2. A trinomial of degree five 25. ͑15x 2 Ϫ 6͒ Ϫ ͑Ϫ8.1x 3 Ϫ 14.7x 2 Ϫ 17͒ 3. A binomial with leading coefficient Ϫ4 26. ͑15.6w Ϫ 14w Ϫ 17.4͒ Ϫ ͑16.9w4 Ϫ 9.2w ϩ 13͒ 4. A monomial of positive degree 27. 3x͑x 2 Ϫ 2x ϩ 1͒ 3 28. y2͑4y2 ϩ 2y Ϫ 3͒ 5. A trinomial with leading coefficient 4 6. A third-degree polynomial with leading coefficient 1 29. Ϫ5z͑3z Ϫ 1͒ 30. ͑Ϫ3x͒͑5x ϩ 2͒ In Exercises 7–10, write a polynomial that fits the 31.͑1 Ϫ x 3͒͑4x͒ 32. Ϫ4x͑3 Ϫ x 3͒ description. (There are many correct answers.) 33.͑2.5x2 ϩ 5͒͑Ϫ3x͒ 34. ͑2 Ϫ 3.5y͒͑4y3͒ 7. A third-degree polynomial with leading coefficient Ϫ2 Ϫ ͑1 ϩ ͒ ͑ Ϫ 3 ͒ 35.2x 8x 3 36. 6y 4 8 y 8. A fifth-degree polynomial with leading coefficient 8 9. A fourth-degree polynomial with a negative leading In Exercises 37–68, multiply or find the special product. coefficient 37.͑x ϩ 3͒͑x ϩ 4͒ 38. ͑x Ϫ 5͒͑x ϩ 10͒ 10. A third-degree trinomial with an even leading coefficient 39.͑3x Ϫ 5͒͑2x ϩ 1͒ 40. ͑7x Ϫ 2͒͑4x Ϫ 3͒ 41.͑2x Ϫ 5y͒2 42. ͑5 Ϫ 8x͒2 In Exercises 11–16, write the polynomial in standard form. Then identify the degree and leading coefficient of the 43. ͑x ϩ 10͒͑x Ϫ 10͒ polynomial. 44. ͑2x ϩ 3͒͑2x Ϫ 3͒ 11.3x ϩ 4x2 ϩ 2 12. x2 Ϫ 4 Ϫ 3x4 45. ͑x ϩ 2y͒͑x Ϫ 2y͒ 13.5 Ϫ x6 14. Ϫ13 ϩ x2 46. ͑4a ϩ 5b͒͑4a Ϫ 5b͒ 15.1 Ϫ x ϩ 6x4 Ϫ 2x5 16. 7 ϩ 8x 47. ͑2r 2 Ϫ 5͒͑2r 2 ϩ 5͒ 48. ͑3a3 Ϫ 4b2͒͑3a3 ϩ 4b2͒ In Exercises 17–20, determine whether the expression is a 49.͑x ϩ 1͒3 50. ͑y Ϫ 4͒3 polynomial. If so, write the polynomial in standard form. 51.͑2x Ϫ y͒3 52. ͑3x ϩ 2y͒3 Ϫ 17.3x ϩ 4x3 Ϫ 5 18. 5x4 Ϫ 2x2 ϩ x 2 ͑1 Ϫ ͒2 ͑3 ϩ ͒2 53.2x 5 54. 5t 4 x2 ϩ 2x Ϫ 3 ͑1 Ϫ ͒͑1 ϩ ͒ ͑ ϩ 1͒͑ Ϫ 1͒ 19.Ίx2 Ϫ x4 20. 55.4x 3 4x 3 56. 2x 6 2x 6 6 333371_0P03.qxp 12/27/06 9:32 AM Page 33

Section P.3 Polynomials and Factoring 33

57.͑2.4x ϩ 3͒2 58. ͑1.8y Ϫ 5͒2 In Exercises 115–120, factor by grouping. 59. ͑Ϫx2 ϩ x Ϫ 5͒͑3x2 ϩ 4x ϩ 1͒ 115. x 3 Ϫ x 2 ϩ 2x Ϫ 2 60. ͑x2 ϩ 3x ϩ 2͒͑2x2 Ϫ x ϩ 4͒ 116. x 3 ϩ 5x 2 Ϫ 5x Ϫ 25 61. ͓͑x ϩ 2z͒ ϩ 5͔͓͑x ϩ 2z͒ Ϫ 5͔ 117. 6x2 ϩ x Ϫ 2 62. ͓͑x Ϫ 3y͒ ϩ z͔͓͑x Ϫ 3y͒ Ϫ z͔ 118. 3x 2 ϩ 10x ϩ 8 63.͓͑x Ϫ 3͒ ϩ y͔2 64. ͓͑x ϩ 1͒ Ϫ y͔2 119. x3 Ϫ 5x2 ϩ x Ϫ 5 65. 5x͑x ϩ 1͒ Ϫ 3x͑x ϩ 1͒ 120. x3 Ϫ x2 ϩ 3x Ϫ 3 66. ͑2x Ϫ 1͒͑x ϩ 3͒ ϩ 3͑x ϩ 3͒ 67. ͑u ϩ 2͒͑u Ϫ 2͒͑u 2 ϩ 4͒ In Exercises 121–152, completely factor the expression. 68. ͑x ϩ y͒͑x Ϫ y͒͑x 2 ϩ y 2͒ 121.x 3 Ϫ 16x 122. 12x 2 Ϫ 48 123.x 3 Ϫ x 2 124. 6x 2 Ϫ 54 In Exercises 69–74, factor out the common factor. 125.x 2 Ϫ 2x ϩ 1 126. 9x 2 Ϫ 6x ϩ 1 69.4x ϩ 16 70. 5y Ϫ 30 127.1 Ϫ 4x ϩ 4x 2 128. 16 Ϫ 6x Ϫ x2 71.2x 3 Ϫ 6x 72. 3z3 Ϫ 6z2 ϩ 9z 129.2x 2 ϩ 4x Ϫ 2x 3 130. 7y 2 ϩ 15y Ϫ 2y3 73. 3x͑x Ϫ 5͒ ϩ 8͑x Ϫ 5͒ 131.9x 2 ϩ 10x ϩ 1 132. 13x ϩ 6 ϩ 5x 2 74. ͑5x Ϫ 4͒2 ϩ ͑5x Ϫ 4͒ 1 2 Ϫ 1 Ϫ 1 1 2 ϩ 2 Ϫ 133.8x 96x 16 134. 81x 9 x 8 135.3x 3 ϩ x 2 ϩ 15x ϩ 5 136. 5 Ϫ x ϩ 5x 2 Ϫ x 3 In Exercises 75–82, factor the difference of two squares. 137. 3u Ϫ 2u2 ϩ 6 Ϫ u3 2 Ϫ 2 Ϫ 75.x 64 76. x 81 138. x 4 Ϫ 4x 3 ϩ x 2 Ϫ 4x 2 Ϫ Ϫ 2 77.48y 27 78. 50 98z 139. 2x3 ϩ x2 Ϫ 8x Ϫ 4 2 Ϫ 1 25 2 Ϫ 79.4x 9 80. 36 y 49 140. 3x3 ϩ x2 Ϫ 27x Ϫ 9 81.͑x Ϫ 1͒2 Ϫ 4 82. 25 Ϫ ͑z ϩ 5͒2 141. ͑x 2 ϩ 1͒ 2 Ϫ 4x 2 142. ͑x2 ϩ 8͒2 Ϫ 36x 2 In Exercises 83–90, factor the perfect square trinomial. 143. 2t 3 Ϫ 16 2 Ϫ ϩ 2 ϩ ϩ 83.x 4x 4 84. x 10x 25 144. 5x 3 ϩ 40 2 ϩ ϩ 1 2 Ϫ 4 ϩ 4 85.x x 4 86. x 3x 9 145. 4x͑2x Ϫ 1͒ ϩ 2͑2x Ϫ 1͒2 2 Ϫ ϩ 2 Ϫ ϩ 87.4x 12x 9 88. 25z 10z 1 146. 5͑3 Ϫ 4x͒2 Ϫ 8͑3 Ϫ 4x͒͑5x Ϫ 1͒ 2 Ϫ 4 ϩ 1 2 Ϫ 3 ϩ 1 89.4x 3x 9 90. 9y 2 y 16 147. 2͑x ϩ 1͒͑x Ϫ 3͒2 Ϫ 3͑x ϩ 1͒2͑x Ϫ 3͒ 148. 7͑3x ϩ 2͒2͑1 Ϫ x͒2 ϩ ͑3x ϩ 2͒͑1 Ϫ x͒3 In Exercises 91–100, factor the sum or difference of cubes. 149. ͑2x ϩ 1͒4͑2͒͑3x Ϫ 1͒͑3͒ ϩ ͑3x Ϫ 1͒2͑4͒͑2x ϩ 1͒3͑2͒ 91.x3 ϩ 64 92. x3 Ϫ 1 150. ͑2x ϩ 5͒3͑4͒͑3x ϩ 2͒3͑3͒ ϩ ͑3x ϩ 2͒4͑3͒͑2x ϩ 5͒2͑2͒ 93.y 3 ϩ 216 94. z3 Ϫ 125 151. ͑x2 ϩ 5͒4͑2͒͑3x Ϫ 1͒͑3͒ ϩ ͑3x Ϫ 1͒2͑4͒͑x2 ϩ 5͒3͑2x͒ 95.x3 Ϫ 8 96. x3 ϩ 8 27 125 152. ͑x2 Ϫ 1͒3͑2͒͑4x ϩ 5͒͑4͒ ϩ ͑4x ϩ 5͒2͑3͒͑x2 Ϫ 1͒2͑2x͒ 97.8x3 Ϫ 1 98. 27x3 ϩ 8 99.͑x ϩ 2͒3 Ϫ y3 100. ͑x Ϫ 3y͒3 Ϫ 8z3 153. Compound Interest After 2 years, an investment of $500 compounded annually at an interest rate r will yield an amount of 500͑1 ϩ r͒2. In Exercises 101–114, factor the trinomial. (a) Write this polynomial in standard form. 101.x2 ϩ x Ϫ 2 102. x2 ϩ 5x ϩ 6 (b) Use a calculator to evaluate the polynomial for the 2 Ϫ ϩ 2 Ϫ Ϫ 103.s 5s 6 104. t t 6 values of r shown in the table. 105.20 Ϫ y Ϫ y 2 106. 24 ϩ 5z Ϫ z2 2 Ϫ ϩ 2 ϩ Ϫ r 1 1 107.3x 5x 2 108. 3x 13x 10 22 % 3% 4% 42 % 5% 2 Ϫ Ϫ 2 Ϫ Ϫ 109.2x x 1 110. 2x x 21 500͑1 ϩ r͒2 111.5x 2 ϩ 26x ϩ 5 112. 8x2 Ϫ 45x Ϫ 18 113.Ϫ5u 2 Ϫ 13u ϩ 6 114. Ϫ6x2 ϩ 23x ϩ 4 (c) What conclusion can you make from the table? 333371_0P03.qxp 12/27/06 9:33 AM Page 34

34 Chapter P Prerequisites

154. Compound Interest After 3 years, an investment of (a) Determine the polynomial that represents the total $1200 compounded annually at an interest rate r will yield stopping distance T. ͑ ϩ ͒3 an amount of 1200 1 r . (b) Use the result of part (a) to estimate the total stopping (a) Write this polynomial in standard form. distance when x ϭ 30, x ϭ 40, and x ϭ 55. (b) Use a calculator to evaluate the polynomial for the (c) Use the bar graph to make a statement about the total values of r shown in the table. stopping distance required for increasing speeds.

1 1 r 2% 3% 3 % 4% 4 % 250 2 2 Reaction time distance 1200͑1 ϩ r͒3 225 Braking distance 200 175 (c) What conclusion can you make from the table? 150 155. Geometry An overnight shipping company is designing 125 a closed box by cutting along the solid lines and folding 100

along the broken lines on the rectangular piece of corru- istance (in feet) 75 D gated cardboard shown in the figure. The length and width 50 of the rectangle are 45 centimeters and 15 centimeters, 25 respectively. Find the volume of the box in terms of x. x Find the volume when x ϭ 3, x ϭ 5, and x ϭ 7. 20 30 40 50 60 Speed (in miles per hour) 45 cm x x Figure for 157 xx x 158. Engineering A uniformly distributed load is placed on

15 cm a one-inch-wide steel beam. When the span of the beam is x x feet and its depth is 6 inches, the safe load S (in pounds) is approximated by 1 15 − 2x (45 − 3x) 2 ϭ ͑ 2 Ϫ ϩ ͒2 S6 0.06x 2.42x 38.71 . 156. Geometry A take-out fast food restaurant is constructing When the depth is 8 inches, the safe load is approximated an open box made by cutting squares out of the corners by of a piece of cardboard that is 18 centimeters by ϭ ͑ 2 Ϫ ϩ ͒2 26 centimeters (see figure). The edge of each cut-out S8 0.08x 3.30x 51.93 . square is x centimeters. Find the volume of the box in (a) Use the bar graph to estimate the difference in the safe ϭ ϭ terms of x. Find the volume when x 1, x 2, and loads for these two beams when the span is 12 feet. x ϭ 3. (b) How does the difference in safe load change as the x span increases?

x S 2

− 18 cm 1600 x 26 − 2x 18 x nds) 1400 6-inch beam

x u x 18 − 2x 1200 8-inch beam 26 cm 26 − 2x 1000 800 157. Stopping Distance The stopping distance of an auto- 600 mobile is the distance traveled during the driver’s reaction 400 200

time plus the distance traveled after the brakes are Safe load (in po x applied. In an experiment, these distances were measured 4 8 12 16 (in feet) when the automobile was traveling at a speed of Span (in feet) x miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time R was R ϭ 1.1x and the braking distance B was B ϭ 0.0475x 2 Ϫ 0.001x ϩ 0.23. 333371_0P03.qxp 12/27/06 9:33 AM Page 35

Section P.3 Polynomials and Factoring 35

Geometric Modeling In Exercises 159–162, match the (d) 1 aa factoring formula with the correct geometric factoring model. [The models are labeled (a), (b), (c), and (d).] For instance, a factoring model for 2x 1 1ͨͧx 1 1ͨ bbͧ ؍ 2x2 1 3x 1 1 is shown in the figure.

1 x x x 1 1 1 xx x a 1 1 1111 b x 1 159. a 2 Ϫ b 2 ϭ ͑a ϩ b͒͑a Ϫ b͒ x x x 1 160. a 2 ϩ 2ab ϩ b 2 ϭ ͑a ϩ b͒ 2 161. a 2 ϩ 2a ϩ 1 ϭ ͑a ϩ 1͒ 2 (a) 1 162. ab ϩ a ϩ b ϩ 1 ϭ ͑a ϩ 1͒͑b ϩ 1͒ aa

Geometric Modeling In Exercises 163–166, draw a geometric factoring model to represent the factorization. aa 163. 3x 2 ϩ 7x ϩ 2 ϭ ͑3x ϩ 1͒͑x ϩ 2͒ 164. x 2 ϩ 4x ϩ 3 ϭ ͑x ϩ 3͒͑x ϩ 1͒ 165. 2x 2 ϩ 7x ϩ 3 ϭ ͑2x ϩ 1͒͑x ϩ 3͒ a 1 2 ϩ ϩ ϭ ͑ ϩ ͒͑ ϩ ͒ 1 1 166. x 3x 2 x 2 x 1 a 1 1 Geometry In Exercises 167–170, write an expression in factored form for the area of the shaded portion of the (b) a figure. a b 167. 168. a a − b r r b b r + 2 (c) b aa 169.xx 170. 8 x x aa xxxx x + 3 18 4 5 5 a b 4 (x + 3) b b a b b 333371_0P03.qxp 12/27/06 9:33 AM Page 36

36 Chapter P Prerequisites

In Exercises 171–176, factor the expression completely. Synthesis 171. x4͑ ͒͑ x ϩ ͒3͑ x͒ ϩ ͑ x ϩ ͒4͑ x3͒ 4 2 1 2 2 1 4 True or False? In Exercises 187–189, determine whether 172. x3͑3͒͑x2 ϩ 1͒2͑2x͒ ϩ ͑x2 ϩ 1͒3͑3x2͒ the statement is true or false. Justify your answer. 173. ͑2x Ϫ 5͒4͑3͒͑5x Ϫ 4͒2͑5͒ ϩ ͑5x Ϫ 4͒3͑4͒͑2x Ϫ 5͒3͑2͒ 187. The product of two binomials is always a second-degree 174. ͑x2 Ϫ 5͒3͑2͒͑4x ϩ 3͒͑4͒ ϩ ͑4x ϩ 3͒2͑3͒͑x2 Ϫ 5͒2͑x2͒ polynomial. ͑5x Ϫ 1͒͑3͒ Ϫ ͑3x ϩ 1͒͑5͒ 175. 188. The difference of two perfect squares can be factored as ͑5x Ϫ 1͒2 the product of conjugate pairs. ͑2x ϩ 3͒͑4͒ Ϫ ͑4x Ϫ 1͒͑2͒ 176. 189. The sum of two perfect squares can be factored as the ͑2x ϩ 3͒2 binomial sum squared.

In Exercises 177–180, find all values of b for which the 190. Exploration Find the degree of the product of two trinomial can be factored with integer coefficients. polynomials of degrees m and n. 191. Exploration Find the degree of the sum of two 177. x 2 ϩ bx Ϫ 15 polynomials of degrees m and n if m < n. 178. x2 ϩ bx Ϫ 12 192. Writing Write a paragraph explaining to a classmate 179. x 2 ϩ bx ϩ 50 why ͑x ϩ y͒2  x2 ϩ y2. 180. x2 ϩ bx ϩ 24 193. Writing Write a paragraph explaining to a classmate why ͑x Ϫ y͒2  x2 Ϫ y2. In Exercises 181–184, find two integer values of c such that 194. Writing Write a paragraph explaining to a classmate a the trinomial can be factored. (There are many correct pattern that can be used to cube a binomial sum. Then use answers.) your pattern to cube the sum ͑x ϩ y͒. 181. 2x 2 ϩ 5x ϩ c 195. Writing Write a paragraph explaining to a classmate a 182. 3x2 Ϫ x ϩ c pattern that can be used to cube a binomial difference. Then use your pattern to cube the difference ͑x Ϫ y͒. 183. 3x 2 Ϫ 10x ϩ c 196. Writing Explain what is meant when it is said that a 184. 2x2 ϩ 9x ϩ c polynomial is in factored form. 185. Geometry The cylindrical shell shown in the figure has 197. Think About It Is ͑3x Ϫ 6͒͑x ϩ 1͒ completely a volume of V ϭ ␲R 2h Ϫ ␲r 2h. factored? Explain. R 198. Error Analysis Describe the error. 9 x 2 Ϫ 9x Ϫ 54 ϭ ͑3x ϩ 6͒͑3x Ϫ 9͒ ϭ 3͑x ϩ 2͒͑x Ϫ 3͒ h 199. Think About It A third-degree polynomial and a fourth- degree polynomial are added. (a) Can the sum be a fourth-degree polynomial? Explain or give an example. r (b) Can the sum be a second-degree polynomial? Explain or give an example. (a) Factor the expression for the volume. (c) Can the sum be a seventh-degree polynomial? Explain (b) From the result of part (a), show that the or give an example. volume is 200. Think About It Must the sum of two second-degree ␲ 2 (average radius)(thickness of the shell)h. polynomials be a second-degree polynomial? If not, give 186. Chemical Reaction The rate of change of an an example. autocatalytic chemical reaction is kQx Ϫ kx 2, where Q is the amount of the original substance,x is the amount of substance formed, and k is a constant of proportionality. Factor the expression.