Stewart J. College Algebra: Concepts and Contexts
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T24 ALGEBRA TOOLKIT A
■
Working with Numbers
15–22 ■ Evaluate each expression.
15. (a) (b) (c)
16. (a) (b) (c)
17. (a) (b) (c)
18. (a) (b) (c)
19. (a) (b) (c)
20. (a) (b) (c)
21. (a) (b) (c)
22. (a) (b) (c)
23–34
■ Simplify the expression and eliminate any negative exponent(s). Assume that all
letters denote positive numbers.
23. 24. 25. 26.
27. 28. 29. 30.
31. 32. 33. 34.
35–48 ■ Simplify the expression and express the answer using rational exponents. Assume
that all letters denote positive numbers.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46.
47. 48. 2
s1s
3
2
3
y1y
1xy
1
4
16xy
2
3
8x
2
1x
2
3
16x
3
2
3
2x
5
2
4
x
7
2
4
x
3
12 2a212
3
a
2
2151
3
x2121
4
x2
2
4
b
3
2b12
6
y
5
212
3
y
2
2
2
5
4u
2
√
6
2
5
8u
3
√
4
2
4
3s
3
t
5
2
4
27st
3
2
3
5y
4
2x
3
2x
5
a
s
2>3
2t
1>6
b
6
a
2q
3>4
r
3>2
b
-4
1u
6
√
3
2
1>3
1x
6
y
4
2
-1>2
a
27
k
3
b
-1>3
a
r
1>2
16
b
1>4
5
5>2
5
1>2
w
4>3
w
1>3
13a
2>3
2
2
14b 2
1>2
y
2>3
y
4>3
x
3>4
x
5>4
A
1
8
B
-2>3
1- 272
-4>3
A
1
32
B
2>5
A
25
64
B
3>2
1- 322
2>5
A
4
9
B
-1>2
A
81
16
B
1>4
A-
1
27
B
1>3
16
1>4
A
1
125
B
1>3
1- 322
1>5
49
1>2
2
4
82
4
32
248
23
27228
2
6
1
64
2
4
2562
4
9
2
5
322
3
- 64264
2
4
1
16
2
4
16216
T25
Algebra Toolkit B
Working with Expressions
B.1 Combining Algebraic Expressions
B.2 Factoring Algebraic Expressions
B.3 Rational Expressions
2
B.1 Combining Algebraic Expressions
■
The Form and Value of an Expression
■
Adding Algebraic Expressions
■
Multiplying Algebraic Expressions
■
Special Product Formulas
■
Combining Polynomials
2
■ The Form and Value of an Expression
An algebraic expression is an arithmetic expression that contains letters, where the
letters stand for numbers. The following are algebraic expressions:
Recall that the letters in these expressions are called variables because they can
“vary” over all real numbers for which the expression is defined.
What do these expressions tell us? We can describe the form of an expression in
words. For example, the first expression says “a number plus 2.” The expression has
a different value when different numbers are assigned to the variable x. For example,
if x is 5, then the value of the expression is .5 + 2 = 7
x + 2
x1x + 3 2
x
x + 2
example
1
The Form and Value of an Expression
Consider the expression .
(a) Describe the expression in words.
(b) Find the value of the expression if x is 5.
x
2
+ 3
2
example
2
The Form and Value of an Expression
Consider the trinomial .
(a) What are the terms of this trinomial expression?
(b) Find the value of the expression if a is 2 and x is .
Solution
(a) The expression has three terms: 5ax, 3a, and .
(b) To find the value of the expression, we replace a by 2 and x by .
Replace a by 2 and x by
Calculate
So if a is 2 and x is , the value of the expression is .
■ NOW TRY EXERCISE 11 ■
- 32- 3
=-32
- 3 5 ax + 3a - 8 = 51221- 3 2+ 3122- 8
- 3
- 8
- 3
5ax + 3a - 8
T26 ALGEBRA TOOLKIT B
■
Working with Expressions
Solution
(a) The expression says “a number squared plus 3 divided by 2.” Notice that the
word description is ambiguous (Do we “add 3 then divide by 2,” or do we add
“three divided by 2”?), whereas the algebraic expression is not ambiguous—
the numerator and denominator are treated as if they are in parentheses.
(b) The value of the expression is obtained by replacing x by 5.
Replace x by 5
Evaluate the numerator
Calculate
So if x is 5, the value of the expression is 14.
■ NOW TRY EXERCISE 9 ■
One of the key features of an algebraic expression is the number of terms it has.
An algebraic expression with two terms is called a binomial, and one with three terms
is called a trinomial. An algebraic expression with one term is called a monomial.
= 14
=
28
2
x
2
+ 3
2
=
5
2
+ 3
2
2
■ Adding Algebraic Expressions
Expressions represent numbers. So we can add, subtract, multiply, and divide ex-
pressions. To add or subtract algebraic expressions, we combine “like terms.” For
example, to add the expressions and , we add like terms
as follows:
The next example illustrates the process further.
= 7ax + 5a + 8
16ax + 7a + 22+ 1ax - 2a + 62= 16ax + ax2+ 17a - 2a2+ 12 + 62
ax - 2a + 66ax + 7a + 2
B.1
■
Combining Algebraic Expressions T27
example
3
Adding Algebraic Expressions
Perform the indicated operation and simplify.
(a)
(b)
Solution
(a) We combine the terms involving cx, the terms involving x, and the constants.
(b) We combine the terms involving and the constants.
■ NOW TRY EXERCISES 15 AND 21 ■
= 3x
2
- 5x + 8
12x
2
- 5x + 62+ 1x
2
+ 22= 12x
2
+ x
2
2- 5x + 16 + 22
x
2
= 10cu + 3u + 4
12cu + 2u + 12+ 18cu + u + 32= 12cu + 8cu2+ 12u + u2+ 11 + 32
12x
2
- 5x + 62+ 1x
2
+ 22
12cu + 2u + 1 2+ 18cu + u + 3 2
2
■ Multiplying Algebraic Expressions
The key to working with expressions is to understand that an algebraic expression (no
matter how complicated) represents a number. So an expression can be substituted for
a letter in another expression. In particular, when using the Distributive Property,
the letters A, B, and C can be replaced by any expressions. For example, if we replace
A by the expression , B by x, and C by y, we get
This process of replacing letters by expressions, called the Principle of Substi-
tution, allows us to find an unlimited variety of true facts about numbers.
1a + b21x + y2= 1a + b2
#
x + 1a + b2
#
y
a + b
A1B + C 2= AB + AC
r
example
4
Multiplying Expressions Using the Distributive Property
Expand the product using the Distributive Property: .
Solution
We view the expression as a single number and distribute it over the terms of
.
Distributive Property, ,
,
Distributive Property (used twice)
Associative Property of Addition
In the last step we removed the parentheses, since by the Associative Property the or-
der of addition doesn’t matter.
■ NOW TRY EXERCISE 25 ■
= ax + bx + ay + by
= 1ax + bx2+ 1ay + by2
C = yB = x
A = 1a + b 2
1a + b21x + y2= 1a + b2x + 1a + b 2y
x + y
a + b
1a + b 21x + y 2
T28 ALGEBRA TOOLKIT B
■
Working with Expressions
example
5
Multiplying Binomials Using FOIL
Expand .
Solution
We use the distributive property (or FOIL).
Distributive Property
FOIL
Combine like terms
■ NOW TRY EXERCISE 33 ■
= 6x
2
- 7x - 5
12x + 1213x - 5 2= 6x
2
- 10x + 3x - 5
12x + 1 213x - 5 2
The acronym FOIL helps us to
remember that the product of two
binomials is the sum of the products
of the First terms, the Outer terms,
the Inner terms, and the Last terms.
From Example 1 we see that to multiply two binomials we multiply each term
in one factor by each term in the other factor and add the results. We can remember
this using the mnemonic FOIL (see the margin).
FOIL
1a + b 21x + y 2= ax + ay + bx + by
c
ccc
c
ccc
2
■ Special Product Formulas
Certain types of products occur so frequently that you should memorize them. You
can verify the following formulas by performing the multiplications.
Special Product Formulas
The key idea in using these formulas (or any other formula in algebra) is the Principle
of Substitution mentioned above: We may substitute any algebraic expression for
any letter in a formula. For example, to find , we use Product Formula 2,
substituting for A and for B, to get
1A + B2
2
= A
2
+ 2AB + B
2
1x
2
+ y
3
2
2
= 1x
2
2
2
+ 21x
2
21y
3
2
+ 1y
3
2
2
y
3
x
2
1x
2
+ y
3
2
2
If A and B are any real numbers or algebraic expressions, then
1. Sum and difference of same terms
2. Square of a sum
3. Square of a difference1A - B 2
2
= A
2
- 2AB + B
2
1A + B 2
2
= A
2
+ 2AB + B
2
1A + B 21A - B 2= A
2
- B
2
example
6
Using the Product Formulas
Expand the expressions using the Product Formulas.
(a) (b)
15x + 2 215x - 2 213x + y 2
2
B.1
■
Combining Algebraic Expressions T29
Solution
(a) We view the expression 3x as a single number and use Product Formula 2.
Product Formula 2, and
Simplify
Notice how the formula applies by simply replacing the letters A and B by the
appropriate expressions:
(b) We view the expression 5x as a single number and use Product Formula 1.
Product Formula 1, and
Simplify
Again, notice how the formula applies by simply replacing the letters A and B
by the appropriate expressions.
■ NOW TRY EXERCISES 39 AND 49 ■
1A + B 21A - B2 = A
2
- B
2
15x + 2215x - 22 = 15x 2
2
- 2
2
= 25x
2
- 4
B = 2A = 5x 15x + 2 215x - 22= 15x 2
2
- 2
2
1A + B2
2
= A
2
+ 2
#
A
#
B + B
2
13x + y2
2
= 13x2
2
+ 2
#
13x 2
#
y + y
2
= 9x
2
+ 6xy + y
2
B = yA = 3x 13x + y 2
2
= 13x2
2
+ 2
#
3x
#
y + y
2
example
7
Using the Special Product Formulas
Find the product.
(a) (b)
Solution
(a) Substituting and in Product Formula 1, we get
(b) If we group together and think of this as one algebraic expression, we
can use Product Formula 1, with and .
Product Formula 1
Product Formula 2
■ NOW TRY EXERCISES 51 AND 55 ■
= x
2
+ 2xy + y
2
- 1
= 1x + y2
2
- 1
2
1x + y - 121x + y + 12= 31x + y2- 1431x + y2+ 14
B = 1A = x + y
x + y
12x - 1y212x + 1y2= 12x 2
2
- 11y2
2
= 4x
2
- y
B = 1yA = 2x
1x + y - 1 21x + y + 1 212x - 1y212x + 1y2
2
■ Combining Polynomials
Certain types of algebraic expressions that occur frequently in algebra have special
names; one of these is a polynomial. A polynomial in the variable x is an expression
T30 ALGEBRA TOOLKIT B
■
Working with Expressions
in which each term is a number times a nonnegative integer power of x. For exam-
ple, the following are polynomials:
The terms of the first polynomial above are , , and 7.
We add and subtract polynomials in the same way we add and subtract any
other expressions: by combining like terms. In the case of polynomials, like terms
are ones in which the variable is raised to the same power. In subtracting polyno-
mials, we have to remember that if a minus sign precedes an expression in paren-
theses, then the sign of every term within the parentheses is changed when we re-
move the parentheses:
- 1b + c2=-b - c
3x4x
2
4x
2
+ 3x + 72x
3
- 5x7x
5
+ 5x
2
- 7
example
8
Adding and Subtracting Polynomials
(a) Find the sum .
(b) Find the difference .
Solution
(a)
Group like terms
Combine like terms
(b)
Distributive Property
Group like terms
Combine like terms
■ NOW TRY EXERCISES 57 AND 59 ■
When we multiply trinomials or other polynomials with more terms, we use the
Distributive Property. It is also helpful to arrange our work in table form. The next
example illustrates both methods.
=-11x
2
+ 9x + 4
= 1x
3
- x
3
2+ 1- 6x
2
- 5x
2
2+ 12x + 7x 2+ 4
= x
3
- 6x
2
+ 2x + 4 - x
3
- 5x
2
+ 7x
1x
3
- 6x
2
+ 2x + 42- 1x
3
+ 5x
2
- 7x2
= 2x
3
- x
2
- 5x + 4
= 1x
3
+ x
3
2+ 1- 6x
2
+ 5x
2
2+ 12x - 7x 2+ 4
1x
3
- 6x
2
+ 2x + 42+ 1x
3
+ 5x
2
- 7x2
1x
3
- 6x
2
+ 2x + 42- 1x
3
+ 5x
2
- 7x2
1x
3
- 6x
2
+ 2x + 42+ 1x
3
+ 5x
2
- 7x2
example
9
Multiplying Polynomials
Find the product .
Solution 1
Using the Distributive Property, we have
Distributive Property
Distributive Property
Laws of Exponents
Combine like terms
= 2x
3
- 7x
2
- 7x + 12
= 12x
3
- 10x
2
+ 8x2+ 13x
2
- 15x + 122
= 12x
#
x
2
- 2x
#
5x + 2x
#
42+ 13
#
x
2
- 3
#
5x + 3
#
42
= 2x1x
2
- 5x + 42+ 3
#
1x
2
- 5x + 42
12x + 3 21x
2
- 5x + 42
12x + 3 21x
2
- 5x + 42
B.1
■
Combining Algebraic Expressions T31
Solution 2
Using table form, we have
First factor
Second factor
Multiply by 3
Multiply by 2x
Add like terms
■ NOW TRY EXERCISE 61 ■
2x
3
-
7x
2
- 7 x + 12
x
2
- 5x + 42x
3
- 10x
2
+
8x
x
2
- 5x + 4 3 x
2
-
15x + 12
2x + 3
x
2
- 5x + 4
B.1 Exercises
CONCEPTS
1. To add expressions, we add _____________ terms. So
_____________.
2. To subtract expressions, we subtract
_____________ terms. So
_____________.
3. Which of the following expressions are polynomials?
(a) (b) (c)
4. Explain how we multiply two binomials, then perform the following multiplication:
_____________.
5. The Special Product Formula for the “square of a sum” is
_____________. So _____________.
6. The Special Product Formula for the “sum and difference of the same terms” is
_____________. So _____________.
7–10
■ An expression is given. Describe the expression in words, and find the value of the
expression when x is 3.
7. (a) (b) 8. (a) (b)
9. (a) (b) 10. (a) (b)
11–14
■ An expression is given.
(a) What are the terms of the expression?
(b) Find the value of the expression if a is , b is 4, x is , and y is 3.
11. 12.
13. 14.
15–22
■ Find the sum or difference.
15. 16.
17. 18.
19. 20.
21. 22. 1- 2bu + 3c√ + 72+ 1bu - c√216ax + 7ay - 2a2- 18ax + 4a2
15by - 2b + 72- 12by - 3b + 2214aw + 3w 2+ 15aw - 2w2
1- 2x
2
- 12+ 14x - 2214x
2
+ 2x2+ 13x
2
- 5x + 62
15 - 3x 2+ 12x - 82112x - 72- 15x - 122
x
2
+ b + 51xy 2
2
+ a
2
axy + b2ax - 10a + 1
- 2- 1
51x + 2215 + x221x + 12
2
x
2
+ 1
x -
2
5
x - 2
5
31x - 523x - 5
15 + x 215 - x 2=1A + B 21A - B 2=
12x + 32
2
=1A + B2
2
=
1x + 221x + 3 2=
x
5
+ 2x
4
+
1
3
x + 3x
3
+
2
x
2x
2
- 3x
12xy + 9b + c + 102- 1xy + b + 6c + 82=
13a + 2b + 42+ 1a - b + 12=
SKILLS
T32 ALGEBRA TOOLKIT B
■
Working with Expressions
23–26
■ Multiply the two expressions using the Distributive Property.
23.
24.
25.
26.
27–34
■ Multiply the algebraic expressions using the FOIL method and simplify.
27. 28.
29. 30.
31. 32.
33. 34.
35–56
■ Multiply the algebraic expressions using a Special Product Formula and simplify.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57–60
■ Find the sum or difference of the polynomials.
57.
58.
59.
60.
61–64
■ Find the product of the polynomials.
61.
62.
63.
64. 11 + 2x21x
2
- 3x + 12
12x - 521x
2
- x + 12
1x + 1212x
2
- x + 12
1x + 221x
2
+ 2x + 32
14x
3
- 2x
2
2- 13x
2
- 2x + 42
13x
6
+ 2x
3
- 62- 12x
3
- 2x - 52
14x
4
+ 3x
2
- 12+ 12x
4
+ 3x
3
- 2x2
12x
3
- 4x
2
+ 3x + 22+ 14x
3
- 6x
2
+ x2
1x + y + z21x + y - z212x + y - 3212x + y + 32
1x + 12 + x
2
221x - 12 + x
2
2211x - 12+ x
2
211x - 12- x
2
2
11a
- b211a + b211x + 2211x - 22
12u + √212u - √21x + 3y21x - 3y2
12y + 5212y - 5 213x - 4213x + 42
1y - 321y + 3 21x + 5 21x - 5 2
12 + y
3
2
2
1x
2
+ 12
2
1r - 2s 2
2
12x + 3y 2
2
1x - 3y 2
2
12u + √2
2
11 - 2y 2
2
13x + 42
2
1x - 22
2
1x + 32
2
17y - 3212y - 1 213x + 5212x - 12
14s - 1212s + 5 213t - 2217t - 4 2
1s + 821s - 2 21r - 321r + 5 2
1y - 121y + 5 21x + 4 21x - 3 2
1a + b 21u + √ + w 2
1w + a + b21u + √
2
1a + b 21x - y 2
1a - b 21x - y 2
B.2
■
Factoring Algebraic Expressions T33
2
B.2 Factoring Algebraic Expressions
EXPANDING
2
■ Factoring Out Common Factors
The easiest type of factoring occurs when the terms have a common factor.
example
1
Factoring Out Common Factors
Factor each expression.
(a)
(b)
Solution
(a) The greatest common factor of the terms and is , so we have
(b) We note that
8, 6, and have the greatest common factor 2
and x have the greatest common factor x
and have the greatest common factor
So the greatest common factor of the three terms in the polynomial is , and
we have
■ NOW TRY EXERCISES 7 AND 13 ■
= 2xy
2
14x
3
+ 3x
2
y - y
2
2
8 x
4
y
2
+ 6x
3
y
3
- 2xy
4
= 12xy
2
214x
3
2+ 12xy
2
213x
2
y2- 12xy
2
21y
2
2
2xy
2
y
2
y
4
y
2
, y
3
,
x
4
, x
3
,
- 2
3x
2
- 6x = 3x1x - 22
3x- 6x3x
2
8x
4
y
2
+ 6x
3
y
3
- 2xy
4
3x
2
- 6x
FACTORING
■
Factoring Out Common Factors
■
Factoring Trinomials
■
Special Factoring Formulas
■
Factoring by Grouping
In
Algebra Toolkit B.1
, we used the Distributive Property to expand algebraic ex-
pressions. We sometimes need to reverse this process (again using the Distributive
Property) by factoring an expression as a product of simpler ones. For example, we
can write
1x - 2 21x + 2 2= x
2
- 4
We say that and are factors of .x
2
- 4x + 2x - 2