336 UNIT 5 • Nonlinear Functions and Equations

IInvest invest iggationation 33 Expanding and Factoring Expanding and FactoringThe standard form of rules for quadratic functions is f(x) = a x 2 + bx + c. But, as you have seen in problems of Investigation 2, rules for quadratics often occur naturally as products of linear expressions. Those factored quadratic expressions reveal useful connections between the functions and their graphs.

For example, the next diagram shows graphs of the functions f(x) = x 2 + 2x - 8 and g(x) = - x 2 + 5x. The rules for those functions can also be expressed as f(x) = (x + 4)(x - 2) and g(x) = -x(x - 5), forms that reveal the x-intercepts of each graph.

yf(x)

g(x)

x-4 -3 -2 -1 1 2 3 4

10

-10

15

-15

5

-5

As you work on the problems of this investigation, look for answers to these questions:

What reasoning can be used to expand products of linear factors into equivalent standard form?

How can standard-form quadratic expressions be written as products of linear factors?

1 The next diagram illustrates a visual strategy for finding products of linear expressions.

x x 1 1 1

x

1

1

a. How does the diagram show that (x + 2)(2x + 3) = 2x2 + 7x + 6?

b. What similar diagram would help to find the expanded form of (x + 3)(x + 1), and what is that expanded form?

Name: ___________________________________Unit 2

Lesson 1Investigation 3

LESSON 1 • Quadratic Functions, Expressions, and Equations 337

2 In earlier work with quadratic expressions like -3x(4x - 5), you have seen how the distributive property can be applied to write the equivalent form -12x2 + 15x. A group of students at Spring Valley High School claimed that by using the distributive property twice, they could expand other factored quadratic expressions. Check the steps in their example below and then apply similar reasoning to expand the expressions in Parts a–f.

(x + 5)(x – 7) = (x + 5)x – (x + 5)7 = (x2 + 5x) – (7x + 35) = x2 – 2x – 35a. (x + 5)(x + 6) b. (x - 3)(x + 9)

c. (x + 10)(x - 10) d. (x - 5)(x + 1)

e. (x + a)(x + b) f. (x + 7)(2x + 3)

3 Look back at your work in Problem 2. Compare the standard-form results to their equivalent factored forms in search of a pattern that you can use as a shortcut in expanding such products. Describe in words the pattern that can be used to produce the expanded forms.

4 The next six expressions have a special form (x + a)2 in which both linear factors are the same. They are called perfect squares. Find an equivalent expanded form for each expression. Remember: (x + a)2 = (x + a)(x + a).

a. (x + 5)2 b. (x - 3)2

c. (x + 7)2 d. (x - 4)2

e. (x + a)2 f. (3x + 2)2

5 Compare the standard-form results to their equivalent factored forms in Problem 4. Find a pattern that you can use as a shortcut in expanding such perfect squares. Describe in words the pattern that can be used to produce the expanded form.

6 Write each of these quadratic expressions in equivalent expanded form.

a. (x + 6)(x - 6) b. (x + 6)(x - 3)

c. (2x + 5)(2x - 5) d. (x - 2.5)(x + 2.5)

e. (8 - x)(8 + x) f. (x - a)(x + a)

7 Look back at your work in Problem 6. Compare the standard-form results to their equivalent factored forms to find a pattern that you can use as a shortcut in expanding products like those in Part a and Parts c–f. Describe the pattern in words.

Factoring Quadratic Expressions In many problems that involve quadratic functions, the function rule occurs naturally in standard form f(x) = ax2 + bx + c. In those cases, it is often helpful to rewrite the rule in equivalent factored form to find the x-intercepts of the graph and then the maximum or minimum point.

338 UNIT 5 • Nonlinear Functions and Equations

For example, the height of a gymnast’s bounce above a trampoline is a function of time after the takeoff bounce. The function might have rule:

h(t) = -16t2 + 24t = -8t(2t - 3)

This information makes it easy to find the time when the gymnast hits the trampoline surface again and when she reaches her maximum height.

h(t)

t

Time into Bounce(in seconds)

Hei

ght a

bove

Tram

polin

e (in

feet

)

0.5 1.0 1.50

2

4

6

8

0

10

h(t) = 0 when -8t(2t - 3) = 0 when -8t = 0 or when 2t - 3 = 0 when t = 0 or when t = 1.5

Maximum value of h(t) is h(0.75) or 9.

In general, it is more difficult to write a quadratic expression like ax2 + bx + c in equivalent factored form than to expand a product of linear factors into equivalent standard form. In fact, it is not always possible to write a factored form for given standard-form quadratic expressions (using only integers as coefficients and constant terms in the factors).

8 To find a factored form for a quadratic expression like x2 + 5x + 6, you have to think backward through the reasoning used to expand products of linear factors.

Suppose that (x + m)(x + n) = x2 + 5x + 6.

a. How is the number 6 related to the integers m and n in the factored form?

b. How is the number 5 related to the integers m and n in the factored form?

c. What is a factored form for x2 + 5x + 6?

9 Find equivalent factored forms for each of these standard-form quadratic expressions.

a. x2 + 7x + 6 b. x2 + 7x + 12

c. x2 + 8x + 12 d. x2 + 13x + 12

e. x2 + 10x + 24 f. x2 + 11x + 24

g. x2 + 9x + 8 h. x2 + 6x

i. x2 + 9x + 18 j. 3x2 + 18x + 24

What general guidelines do you see for factoring expressions like these?

LESSON 1 • Quadratic Functions, Expressions, and Equations 339

10 When a quadratic expression involves differences, finding possible factors requires thinking about products and sums involving negative numbers. Write these expressions in equivalent forms as products of linear expressions.

a. x2 - 7x + 12 b. x2 + 5x - 6

c. x2 - 8x + 12 d. x2 - x - 12

e. x2 - 10x + 24 f. x2 + 10x - 24

g. x2 - 9x + 8 h. x2 - 4x

i. x2 - 7x - 18 j. 2x2 + 13x - 7

What general guidelines do you see for factoring expressions like these?

11 The examples here involve some of the special cases you studied in the practice of expanding products. Where possible, write each given expression as the product of linear expressions.

a. x2 - 9 b. x2 - 81

c. x2 + 16 d. x2 + 10x + 25

e. x2 - 6x + 9 f. x2 + 16x + 64

g. 4x2 - 49 h. 9x2 + 6x + 1

What general guidelines do you see for factoring expressions like these? The two special forms are called difference of squares and perfect square quadratic expressions.

Using Computer Algebra Tools When the factoring task involves an expression like -10x2 + 240x - 950, things get more challenging. Fortunately, what is known about factoring quadratic expressions has been converted into routines for computer algebra systems. Thus, some simple commands will produce the desired factored forms and the insight that comes with them.

For example, the next screen display shows how a CAS would produce a factored form of -10x2 + 240x - 950.

12 Use a computer algebra system to write each expression in equivalent form as products of linear factors. Then use your own reasoning to check the accuracy of the CAS results.

a. x2 + 2x - 24 b. x2 - 6x + 5

c. -x2 + 8x - 15 d. 2x2 - 7x - 4

e. 2x2 + 15x + 18 f. 3x2 - 7x - 6

g. -3x2 + 8x h. 5x + 3x2

CPMP-Tools