Download Operations With Polynomials

Chapter Focus
In this chapter we will learn to perform operations [+, -, x, -:-] with polynomials. In the process, we will
apply the rules of exponents for positive and negative integers in order to rewrite and simplify algebraic
expressions. We will also use our skills with polynomial operations in working with word problems.
Chapter Content
Lessons
Standards
6.1
A-SSE.la; A-SSE.lb; A-APR.l
Adding and Subtracting Polynomials
Adding or Subtracting Monomials with Like Terms
Standard Form of Polynomials
Adding and Subtracting Polynomials
6.2
Multiplying a Monomial by a Monomial
A-SSE.2; A-APR.l
Multiplying Powers with Like Bases
Multiplying a Monomial by a Monomial
6.3
Multiplying a Polynomial by a Monomial
A-SSE.2; A-APR.l
6.4
Multiplying a Polynomial by a Polynomial
A-SSE.2; A-APR.l
Using the Distributive Property for Multiplying Polynomials
Using the FOIL Method for Multiplying Polynomials
Using the Box Method for Multiplying Polynomials
6.5
Special Products of Binomials
A-SSE.2; A-APR.l
6.6
Negative Integers as Exponents
N-RN.l; N-RN.2
6.7
Dividing Polynomials
A-SSE.2
Dividing a Monomial by a Monomial
Dividing a Polynomial by a Monomial
CHAPTER 6 REVIEW
Cumulative Review for Chapters 1-6
Chapter Vocabulary
binomial
monomial
square of a binomial
box method
perfect square trinomial
standard form of a polynomial
difference of two squares
polynomial
trinomial
FOIL method
LESSON 6.1
6.1 Adding and Subtracting Polynomials
A term is an algebraic expression written as the product or quotient of numbers,
variables, or both. A term that has no variables is often called a constant.
4x2y
• 5, X, cd, 6mx, - 33 are all terms. Of these, 5 is a constant.
- m
An algebraic expression of exactly one term is called a monomial.
• Examples of monomials include 7, a, and 2x2 •
Like terms contain the same variables with corresponding variables having the
same exponents. Terms are separated by plus ( + ) and minus (- ) signs.
• 7X
3
y4 and X 3y4 are like terms; 5 and 100 are like terms. y and x are not like
terms; nand n3 are not like terms.
Since algebraic expressions themselves represent numbers, they can be added,
subtracted, multiplied, and divided. When algebraic expressions are added or
subtracted, they can be combined only if they have like terms.
Adding or Subtracting Monomials with Like Terms
A monomial is an expression of the form ax n, where a represents a real number and
n represents a positive integer. The real number a is the coefficient, and the number
n is called the degree of the monomial.
For example:
• 7x 2 has a coefficient of 7 and degree of 2.
• -9m has a coefficient of -9 and degree of 1.
• k has a coefficient of 1 and degree of 1.
• 24 has a degree of 0, since 24 = 24xo.
A monomial with
a degree of 0 is a
constant.
Adding and Subtracting Polynomials
211
If more than one variable appears in a term, such as 6X 2y3, the degree of the
monomial is the number of variable factors. The monomial 6x 2y3 can be written as
6·x.x.y.y.y, so it has five variable factors and its degree is 5.
For example:
• 8x 4y3 is a monomial that has a coefficient of 8 and degree of 4
• Sxm 3 is a monomial that has a coefficient of 5 and degree of 1
+ 3 or 7.
+ 3 or 4.
To find the degree
of a multivariable
term, sum the
exponents of the
variables.
To Add or Subtract Monomials with Like Terms
• Use the distributive property and the rules of signed numbers to add or
subtract the coefficients of each term.
• Write this sum with the variable part from the terms.
e EIiII
Show how to add -z.x3 and Sx3 using the distributive property.
SOLUTION
-2x3 + Sf
e EIiII
= (-2 + S)x3 = 3x3
Show how to subtract 7mn 2 from 4mn2 using the distributive property.
SOLUTION
4mn2
-
7mn 2 = (4 - 7)mn 2 = -3mn 2
Standard Form of Polynomials
An algebraic expression of one or more unlike terms is a polynomial.
Binomials are polynomials with two unlike terms. 7v + 9 and 3x2 - 8y are both
binomials. Trinomials are polynomials with three unlike terms. x 2 - 3x - 5 and
3a2 bx - Sax - 2ab are both trinomials.
A polynomial with one variable is said to be in standard form when it has
no like terms and is written in order of descending exponents. For example,
4x + 9 - Sx2 + 3x3 in standard form is 3x3 - 5x 2 + 4x + 9.
To Write a Polynomial in Standard Form
• Combine like terms.
• Arrange the terms in order of descending exponents. The "+" and"
signs
belong to the coefficient to the right of the sign. When you are arranging the
terms, treat the terms as positive or negative based on the sign in front of the
term.
_II
The degree of a polynomial is the degree of the term with the greatest degree.
For example:
• -2xs + 3x3 + 6x 2 + 8 is a polynomial of degree 5, taken from xS•
+ a3b4
+ ab s is a polynomial with two variables, a and b. The degree
of the first term is 4, the second term is 3 + 4 or 7, and the third term is
1 + 5 or 6. The degree of the polynomial is 7.
• a4
212
Chapter 6: Operations with Polynomials
When you are
asked to simplify
a polynomial, you
should always
write it in
standard form.
.'
•i
I
;..
['t
e GIiIJ
Given the polynomial 6x +
2.x2 - (11 + 3x2)
a Write the polynomial in standard form.
b Identify the degree of each term.
c Identify the degree of the polynomial.
SOLUTION
-x 2 + 6x - 11
a
b The degree of - x 2 is 2, 6x is 1, and -11 is O.
2
c
a Write the polynomial in standard form.
If there is more
than one variable
in the polynomial,
arrange the terms
in descending
order of one of
the variables.
b Identify the degree of each term.
c Identify the degree of the polynomial.
SOLUTION
a 4a 3b - 2a 2b2 + 3ab3 + 9
b The degree of 4a 3b is 4, -2a2b2 is 4, 3ab3 is 4, and 9 is O.
c 4
PRACTICE
mB Exercises 1-10: In each of the following,
4. 6y + 4y 3 + 11 - 2y 2
a
b
c
d
5. 7 - 2.x - 8x 3 + 10x2
Write the polynomial in standard form.
Name the coefficient of each term.
State the degree of each term.
State the degree of the polynomial.
+ 3x + 3x 2 + 1
1.
x3
2.
2a 3
3.
5x - 4x 3
-
3 - 6a 2 + 5a
+ 15x2 + 10
6. a3
-
ab 3 + 3a 4b2
-
15x 4
-
3a 2b
7. 3xy + y4 - X2y 2 + x 3if
8. 5X 3y 2 + 8xy - 6 + 4x 2y
9. 2x4y - X3y 2 + X2y 3
10. -abc + 7 + 5a 5 b4 c3
xy4
-
+1
2a3 b2c
Adding and Subtracting Polynomials
213
Adding and Subtracting Polynomials
To Add Polynomials
• Use the commutative property to rearrange the terms so like terms are
beside each other.
• Combine like terms.
Remember, a - b
is the same as
To Subtract Polynomials
a + (-b).
• Change the sign of every term in the subtracted polynomial and remove
parentheses.
• Combine like terms.
1. Add -3x2 + 4y and 5x3
-
6x2
3y.
-
SOLUTION
(-3x 2 + 4y) + (5x 3
-
6x 2
-
3y)
= - 3x2 + 4y + 5x3 - 6x 2 - 3y
= 5x 3
-
3x 2
6x2 + 4y - 3y
-
= 5x3 + (-3 - 6)x2 + (4 - 3)y
= 5x 3
-
9x 2 + 1Y
= 5x 3
-
9x 2 + Y
2. Subtract 9x 2
5x from -4x 2
-
-
8x.
SOLUTION
First, rewrite the problem to show the subtraction
as ( -4x2 - 8x) - (9x 2 - 5x). Change the signs in
the subtracted polynomial and remove the parentheses.
(-4x 2
= -4x2
=
8x) - (9x 2
-
-
Check. (-13x 2
= -13x2
3x + 9x 2
-
= (-13x 2 + 9x2 ) +
5x)
= -4x2
8x - 9x 2 + 5x
-
3x) + (9x 2
-
-
-
5x)
5x
(-3x - 5x)
8x .I
(-4 - 9)x2 + (-8 + 5)x
= -13x 2
o El~.1
-
3x
Write the missing length as an expression and show your work.
1•
7x-4Y-
71
1 ? 1_3x+ 8Y
16x+9y
•
SOLUTION
Subtract the sum of the two shorter lengths from the entire length. 16x + 9y - [(7x - 4y) + (3x + 8y)]
Use the distributive property and simplify.
16x + 9y - 7x + 4y - 3x - By
=6x+5y
214
Chapter 6: Operations with Polynomials
PRACTICE
1. Simplify (7x + 6x) - 12x.
A. x
B. 2x
C. X 2
D. 1
-
9) - (- 2x2
X -
17. (x 3
-4
-
3x2 + 2x + 5) - (- 5x 3 + x2 + 3x - 2)
18. What polynomial will produce the sum
6x 2 - x + 2 when added to 4x 2 - 6x + 3?
IUIiII] Exercises 19-26: Use the distributive
0
property, remove the parentheses, and solve each
of the following equations.
2m - 2
2m
3. Find the sum of - 3x2 - 4xy + 2y 2 and
-x2 + 5xy - 8y 2.
A. -2x2 + xy - 6y 2
-4x + xy - 6f
C. -4x2 - 9xy - 6f
D. -2x2 - 9xy + 10y2
19. 13x - (x
+ 21)
39
= 35
20. 3a - (5 - 2a)
21. 7x - (x 2 -
=
X -
9) = 17 - x 2
2
B.
4. What is the result when -4x + 6 is subtracted
from 8x + 6?
22. 19m - (l - 2m - m2) = 20 + m2
23. 10 - (x + 6) - 2 = -5x + 9 - (-5x + 3)
24. O.5a + (a - 1) - (0.2a + 10) = 0.2a - (0.2a - 28)
25. 2x - [5x - (6x + 2)] = 14 - x
A. 12x + 12
B. 12x
C. 4x+6
D. 4x
26. 7x - [x - (2x + 8)] = 3x - 7
5. Simplify 5x - 3y - 7x + y.
6. What is the result when lOx - 7 is subtracted
from 9x - IS?
27. What polynomial must be added to
a2 - 3a + 8 to obtain the sum 3a2 + Sa - 9?
28.
e:IiIlIJ The perimeter of the given
2
isosceles trapezoid, ABCD, is 20x + 18.
Base AD = 8x 2 + 1 and base BC = 4x 2 + 3.
7. What is the result when 3 - 2x is subtracted
from the sum of x + 3 and 5 - x?
8. Add 2x - 3x2 - 7, 3 - 5x - 5x 2, and
2x2 + 12 + x - x2.
9. Find the sum of 4x2
3x2 - 5x + 7.
-
A
6x - 3 and
Exercises 11-17: Remove parentheses and find
each sum or difference.
11. (x 2 + 3x + 1) + (2x 2 - X - 2)
12. (- 3x2 + 4x - 8) + (4x 2 + 5x - 11)
-
4)
+
(x 2
-
X
+ 4)
U
8x2+1
D
a What is the length of one of the legs, either
AB or DC?
b Let x = 2. Substitute and check if your
answer is plausible.
10. From 16x 2 + 25y + 12z subtract
16x 2 - 5y + 8z.
13. (4x 2
+ X + 4)
16. (x2+2x-3) - (x 2 -4)
2. Simplify m - [2 - (2 - m)].
A.
B.
C.
D.
15. (x 2
29.
e:IiIlIJ The measures of2two sides of a
triangle are 8x + 8 and 2x + 7.
a If the perimeter, P, measures
10x2 - 3x + 12, what is the measure of the
third side?
b Let x = 3 and show that your answer makes
sense.
14. (- 3x2 + 6x + 1) - (4x 2 + 7x - 3)
Adding and Subtracting Polynomials
215
LESSON 6.2
6.2 Multiplying a Monomial by a Monomial
Multiplying Powers with Like Bases
When multiplying powers with like bases, count the number of times that the base
is used as a factor or simply add the exponents.
The following is a list of guidelines for working with exponents.
Rules for Operations on Terms with Exponents
Operation
Addition and subtraction
x" + x m = x" + x m
x" -xm =x" -xm
Multiplication
m
x" • x =x
,,+ m
Rule
Like bases with unlike
exponents cannot be added
or subtracted unless they
can be evaluated first.
To multiply powers of like
bases, add the exponents.
Examples
3
2 + 2 = 4 + 8 = 12
3
3 - 32 = 27 - 9 = 18
2
i + a3 = a2 + a3
2
3
2
3
a-a=a-a
4
3
X
35 = 39
a2 • a3 = i + 3 = i
a-4 • i = a-4 + 5 = a1 = a
7
Division
X"
"-m
---,;;-=x
,xi' 0
x
Raising a power to a
power
(x m )" = XmIl
Raising a fraction to a
power
(~r = ~:
Raising a product to a
power
(xy)" = x"y"
To divide powers of like
bases, subtract the exponent
of the divisor from the
exponent of the dividend.
To raise a term with an
exponent to some power,
multiply the exponents.
To raise a fraction to a
power, raise the numerator
and the denominator to that
power.
To raise a product to a
power, raise each factor to
that power.
SOLUTION
105 + 6 = lOll
2.
y106 • y14
3. b • b4
216
•
blO
y106 + 14
= y120
b1 + 4 + 10 = blS
Chapter 6: Operations with Polynomials
7
±---4
5 4
5
2
-4
-
5
a
5­ 2
= a3
z=a
a
3
a
3- 8
-5
1
s=a
=a =5
a
a
(5 2)3 = 52 x 3 = 56
(a 4 )3 = a4 • 3 = a12
(~r = ~: = ;5
(~r = ::
(5 • 2)3 = 53 • 23 = 1,000
(4a)2 = 42 • i = 16i
(ab)5 = a5b5
To Find Powers of a Power
•
•
•
•
Raise the numerical coefficient to the indicated power.
Follow the rule for signs in multiplication.
Multiply the exponents of the given literal factors by the power.
If any given term is a fraction, raise both the numerator and the denomina­
tor to the indicated power.
IE~~:]Model Problems 1-9: Simplify and show your work.
lU I
(3a2xy3)2
(3a2xy3)(3a2xy3) = (3)(3)(a2)(a2)(x)(x)(y3)(y3) = 9a4x2y6
( -3xy4)2
(-3xy4)( -3xy4) = (-3)( -3)(X)(X)(y4)(y4) = 9xly 8
(4bx2)3
(4bxl)(4bx2)(4bxl) = (4)(4)(4)(b)(b)(b)(xl)(xl)(xl)
(-3m 4 )3
( -22x)4
= -27m 12
(-2)4[(2)x]4 = (-2)4(2)4(X 4) = ( -2 )4C12X4 =
( -2xla)5
(-2)5x lOa5 = -32xlOa5
= 43b3x6 = 64~X6
(-3)3m 12
16c12X4
(~ a b Y
2 3
(O.3a 3x5)2
PRACTICE
Exercises 1-20: Simplify.
12. (-3x2y4)5
1. (3ab 3 )2
2. ( -3x 2y 3)2
3. (am4x )3
4. ( -x 3 )3
13.
14. (-0.3x 2)3
7. ( -Sx 4)2
y
15.
(~ ab3x2
16.
(~)2
3
5. ( -2m 3x)4
6. (Sa 3b1O )2
(~ a4bSY
-Sc
2y4
8. ( -3a2xS)3
17. ( -3X2
2a
9. (4aSx4 )3
18. (abc)X
10. (2x 4y 6)S
19. (aXbx)m
11. (-0.5ab 4)2
20. (6x2yb)a
Y
Multiplying a Monomial by a Monomial 217
Multiplying a Monomial by a Monomial
To Find the Product of Monomials
• Multiply the numerical coefficients using the rule of signs for multiplication.
• Multiply variables of the same base by adding the exponents.
• Multiply these two products together.
~ Model Problems 1-2: Simplify and show your work.
o
Multiply -4a2b3 and 3a3b.
SOLUTION
(-Wb 3 )(3a3b)
= (-4
• 3 )(a2
•
a3
b3
•
•
b1 )
= -12asb4
o
Simplify (3x")2(4x b).
SOLUTION
(3x")2(4i') = 32x 2a
•
4xb = (9. 4)(? • xb) = 36?+b
PRACTICE
Exercises 1-25: Simplify.
1. Sx2
•
2x4
2. (- 3a 3 )
( -
14. (8a) (-2a X )
4x2 )
3. (Sa 3) (-4a 7 )
4. (-8r4 ) (2?)
5. (- 3ab 2? ) (2a 2bc3 )
6. (-4x Sy) (x 7y)
7.
(-~)(-1)
8. (2xb 2)( Sb 4)
9. -ab(a 3b ) (an )(a4n )
10. - Sx lO (20x 20 )
11. -6a3b2(8a 2bS)
12.
(~
13. (-X)3( -2x)5
x 4yl ) (2x 3yS )
15. (21)( -7y&)
16. (Sa 3 )(2a2 )3
17. ( -x2) (3xyl)
18. (San )2 (3xyl )
19. (-4m l +2) (Sm 21 )2
20. (-xYZ)2( -4x4)2
21. (3xy )3( - 2x2)3 ( - x 2y )3
22. ( 7X2y 3z 4) 2 ( - 2x4y3z 2) 3
23.
(~a2b3Y(- ~a4bS
y
24. (Sx 2y 3z ) (2xy2 z 4) ( - xyz )
25. 2am( -mr)3( -3a2r2)
Practice Problems continue . ..
218
Chapter 6: Operations with Polynomials
Practice Problems continued . ..
Exercises 26-30: Rewrite the expressions as a
monomial. Show your work.
26. (- 2x 2y)3 + (3x 3 )( Sxy )3
32.
27. (-2a) ( -3ab)3 + (3a )2(ab)2( -6b)
33.
28. (2x)2( -2Sx4 ) + (-4x 4 ) (-x)2 + (-10x 3 )2
29. xQ(x b ) (xc) + 3xQx bx c - 2xQx bx c
34.
2
and
2
,
the width is 7ab what is the area of the
rectangle?
Il!LIiIJ Find the area of a square if a side is
12k.
Il!LIiIJ What is the volume of a rectangular
solid with a length of lOn, a width of 3n, and
30. (-3ax 2) (3ax 3 )(ax)2 + (-2ax)2(ax 4 )(4ax)
31.
Il!LIiIJ If the length of a rectangle is Sa
a height of
Il!LIiIJ Represent the distance an airplane
35.
flies in 8 hours at the rate of 7Sx miles an
hour.
in?
Il!LIiIJ What is the cost of Sm dozen golf
balls at 3m 2 dollars per dozen?
LESSON 6.3
6.3 MUltiplying a Polynomial by a Monomial
To Find the Product of a Polynomial and a Monomial
• First use the distributive property, where a(b + c) = ab + ac, to remove
parentheses.
• After multiplying each term of the polynomial by the monomial, simplify
the product sums, if possible.
mill Model Problems 1-4: Simplify and explain your reasoning.
e
Simplify -7x(x 2
-
2) - 9x.
SOLUTION
Write the expression.
-7x(x2
2
-
2) - 9x
Use the distributive property.
-7x(x
Multiply.
-7x3 + 14x - 9x
Add like terms.
-7x3 + 5x
) -
7x( -2) ­ 9x
SOLUTION
Use the distributive property.
~ r2x(8~) + ±r2x(12rx) +±~x( -4x2)
Multiply.
Model Problems continue . ..
Multiplying a Polynomial by a Monomial
219
Model Problems continued
The quantity (5x + x + 4) is
immediately preceded by
the coefficient -3x and not
by the term 5x2 •
Simplify 5x2 - 3x(5x + x + 4) + (2x)2.
SOLUTION
5x 2 - 3x(6x + 4) + (2X)2 '------""'!!'I!~I!!T~~U
Simplify in the parentheses first.
Remove parentheses using distributive property.
5x2 - 3x(6x + 4) + 4x 2
5x 2 - 18x2 - 12x + 4x2
Combine like terms.
-9x2
Apply the exponent in the last term.
,
-
12x
Simplify b2 - 4[2b - 3(b - 5)].
SOLUTION
Simplify in the inner parentheses first.
b2 - 4[2b - 3b + 15]
Simplify within the brackets.
b2 - 4[ -b + 15]
Remove brackets using distributive property.
b2 + 4b - 60
PRACTICE
Exercises 1-24: Simplify.
1. 5(3b + 1)
2. 8(3x - 4)
3. -3(5w + 6)
4. -4( -5x - 1)
16. 2'1 (3 y b - y3)
17.
~b(4b2-8b+16)
18. a(aX -3)
19. x· (5x 3•
-
x)
5. 7 ( - 2x2 - 4x)
6. x 2(3x 3
-
5)
7. 5x(3x2 + 2)
8. a(a 2 - 2ab + b2)
21. xy(3x 2 + 6xy - 4 y 2)
22. -5a 3 (2a 2 - 3a - 1)
23. 2x2Y(X 2 - 3xy - 2f)
9. -x2(5x 2 - X + 3)
10. -3a2(a 2 - 6a + 9)
11. -3x4 ( -4x2 - 3x + 2)
12. ax (6x - a)
13. 4xy(5x2 - 7y2)
14. -x ( - x 2 - x)
Elm
Exercises 25-29: Rewrite the
following products as equivalent
polynomials.
25. a2b(a3b - 4a 2b2 - 5ab3 )
26. -4a 2x (5a 2 - ax - 5x 2)
Practice Problems continue . ..
220
Chapter 6: Operations with Polynomials
Practice Problems continued . ..
29. 0.4a(0.5a 2
-
O.3a - O.ltt)
30. Multiply
x5
+ 2x4
-
3x 3
+ x2 -
X
+ 1 by 3x2•
Exercises 31-50: Simplify.
31. -5(4x - 3) + 6x - 4
32. 4 + 7 (m - 3) + 3m
33. 9 - 4 (3c + 10) - 10
34. Bb - b(2b - 1)
35. 4x - 3x(5x - 6) + Br
36. 9x + 4 - (2x - 6)
37. 4x2
-
3x - (4x - 9r)
+ (3x)2
38. -a - 4[2a - (3x - 5)]
39. x + 3(x - 2) - (x - 1) - 1
40. x - (x 2
-
X -
6) - (x 2
-
6x
+ 5)
41. 2x - [3x + B + 2(3 - 5x)]
42. 3x + [2x - 3 (5 - 4x)]
43. 2 + 3[5x - 3(4y + x)]
44. -7x [5 (2x - y) - (3x + 2y)]
45. -2[5 + 2(2x - 1)]
46. 2x - [- 3x - (3x + 4) + 5]
47.5[3a-4(a-2)]
48. ~ [2a - (x - 1)a]
49. 2a - [-2(x - l(a - 1) - 1)]
50. 5 - x [x - x [ (x - 1) - x] - x]
Multiplying a Polynomial by a Monomial
221
LESSON 6.4
6.4 Multiplying a Polynomial by a Polynomial
When multiplying polynomials, each term of one polynomial must multiply each
term of the other polynomial. The following are three common methods for mul­
tiplying polynomials:
1. Distributive property (horizontal method)
2. FOIL method (used only to multiply two binomials)
3. Box method (rectangular area)
All the methods use the distributive property. After using anyone of these meth­
ods, combine like terms.
Using the Distributive Property for Multiplying
Polynomials
• Distribute the first polynomial over the terms of the second.
• Solve as before.
1. (x+2)(x+3)
SOLUTION
Given.
Distribute (x + 2) over x and 3.
Distribute x over the first (x + 2) and 3 over the second.
Combine like terms.
o
(2x - 3) (3x 2
= (2x = 6x 3 = 6x3 -
-
3) 3x 2
9x
2
5x
+
(2x - 3) ( - 5x)
2
-
10x
2
+ 23x
19x
+ 4)
+ 15x + 8x
+
(2x - 3) 4
- 12
- 12
3. (x + 4)(2x - 3)
SOLUTION
+ 4)(2x - 3)
= x(2x) + x( -3) + 4(2x) + 4( -3)
= 2x 2 - 3x + 8x - 12
= 2x2 + 5x - 12
(x
222
Chapter 6: Operations with Polynomials
(x+2)(x+3)
= (x + 2)x +
(x
+ 2)3
2
= x + 2x + 3x + 6
= x 2 + 5x + 6
The distributive property
can also be used this way:
Distribute the first term to
each term in the second
parentheses and then the
second term to each term
in the second parentheses.
'I,
Using the FOIL Method for Multiplying Polynomials
Use the FOIL method to multiply (2x + 1)(3x - 1).
F
Multiply the first terms. (2x. 3x) = 6x 2
o
Multiply the outer terms. (2x)· (-'-1)
I
Multiply the inner terms. (1· 3x) = 3x
L
Multiply the last terms. 1 • (-1) = -1
FOIL stands
for First, Outer,
Inner, Last.
= -2x
tFfl
+
(2x
1)
(3x - 1)
'(3X/
-2x
Next, combine like terms, so 6x2 + (-2x) + 3x + (-1) = 6x2 +
X -
1.
1. (x+3)(x+4)
2. (2a - 3)(Sa + 2)
SOLUTION
SOLUTION
tFll
+ 3)
+ 4)
(x
(x
'(3X/
n====1l
+ 2)
(2a - 3)
\~lSj)
4x
x
2
product of
first terms
x- x
+
7x
+4a
+ 12
sum of
inner and
outer
3x
(Sa
10a 2
-
lla -
6
product of
last terms
3-4
+ 4x
Using the Box Method for Multiplying Polynomials
To Use the Box Method
• Make a table of products.
• List the areas of the four inside rectangles, and then combine to find the total
area of the whole rectangle.
Multiplying a Polynomial by a Polynomial
223
1. Multiply (x + 1) by (3x + 2).
+
3x
SOLUTION
x
2
2x
3x2
+1----------+ --------1
2
3x
Combine the items in the boxes: 3x 2
Answer:
+ 2x + 3x + 2 = 3x2 + 5x + 2
3x 2 + 5x + 2
2. Multiply (x + 2) (2x 2 +
X -
3).
x
-3
2x3
x2
-3x
4x 2
2x
-6
+
SOLUTION
x
+
2
Combine the items in the boxes: 2x 3 + x2
Answer:
2x 3
+ 5x2 -
X -
-
3x
+ 4x2 + 2x - 6 = 2x 3 + 5x 2 -
6
X -
6
PRACTICE
mill Exercises 1-25: Find the product.
13. (2x + 3d) (4x + d)
1. (x+4)(x-3)
14. (x 2 + x)(x + 1)
2. (a + 5) (a + 6)
15. (3x 2 + 2)( x + 5)
3. (b - 7) (b - 4)
16. (a + b)(a + x)
4. (x + 3)(x + 3)
17. (2x 2 + 5)(x - 1)
5. (x + 8)2
18. (x - 1 )(x2
6. (c - 2)2
19. (8ab + x)( 3ab + x)
7. (x + y)(x + y)
20. (2x 2
8. (a - b)( a - b)
21. (x + 1 )(x2 + 2x + 1)
9. (2x - 6)(4x + 3)
22. (x + 3)3
-
2x
-
+ 1)
3x)(x - 1)
10. (3x + 5)( x - 1)
23. (a - b)(a2 + 2ab + b2 )
11. (2x + 3)(5x + 5)
24. (x + 3)(5x 2 + 3x - 2)
12. (3ax - 2)( ax - 5)
25. (2x - 5 )(x3
-
2x 2
-
X
+ 3)
Practice Problems continue . ..
224
Chapter 6: Operations with Polynomials
Practice Problems continued . ..
ELIiII Exercises
32. a If the sides of a rectangle are 4x + 9 and
x - 8, express the area of the rectangle as
a polynomial.
26-30: Simplify. Identify the
method you used to simplify the expression.
Show your work.
b Let x = 10. Show that the expressions for
length and area are consistent.
26. x(x - 2) + (x + 2)2
27. 7a 3 + a(a - 3)2
33. a If the edge of a cube is x - I, express the
volume of the cube as a polynomial.
28. lOx 2 - (3x + 1 )(3x - 1)
29. 5m 2
-
b Let x = 3. Show that the expressions for
length and volume are consistent.
2(m + l)(m + 1) + 4
30. (x + 1)2 - (x - 1)2
34. a If three consecutive integers are repre­
sented as x, x + I, and x + 2, write the
product of those three integers as a poly­
nomial in simplest form.
EIB
Exercises 31-35: Write an algebraic
expression that represents the answer. Express the
answer as a polynomial in simplest form.
b If x = 2, determine the three consecutive
integers.
31. a If the length of a square measures
3x - 2, express the area of the square as a
polynomial.
35. A truck travels at the average rate of
lOx + 25 miles per hour.
b Let x = 2. Show that the expressions for
length and area are consistent.
a Represent the distance traveled in x + 5
hours.
b If x = 4, how fast does the truck travel?
c If x = 4, how far does the truck travel in
x + 5 hours?
LESSON 6.5
6.5 Special Products of Binomials
Certain binomial products appear so often that you should know them on Sight.
You should be able to recognize the structure of the binomial and easily simplify it.
EIiIIl Model Problems 1-5: Use the struc­
ture of the binomial product to simplify. Show
your work.
o
(m - 4)(m
+ 4)
SOLUTION
(mf - (4f
=
m2 - 16
o
(3y
+ 4f
SOLUTION
(3y + 4)(3y + 4) = (3y)2
= 9y2 + 24y + 16
o
(5x - 3f
SOLUTION
(5x - 3)(5x - 3) = (5X)2
= 25x2 - 30x + 9
SOLUTION
+ 2(3y)(4) + 42
e
(a - b)(a
+ 2(5x)( -3) + (-3f
+ b)(a2 - b2)
SOLUTION
(a 2 - b2)(a 2 - b2) = (a 2f
= a4 - 2a 2b2 + b4
+ 2(a 2)( -b2) + (b2)2
Special Products of Binomials 225
Product of a Sum and a Difference:
Distributive Method
2
= x - xy
- y)
= X 2 _ y2
Box Method
Find the product: (x + y)(x - y)
E Multiply the first terms:
2
(x)(x) = x
X
o Multiply the outer terms:
(x)( -y)
= -xy
x
I Multiply the inner terms:
(y)(x) = yx
+ y)(x - y)
= x(x - y)
+ y)(x
FOIL Method
Distribute the first term, x, to
each term in the second paren­
theses. Then distribute the second
term, y, to each term in the sec­
ond parentheses.
(x
(x
(y)( _y)
+ yx - Y2
Y
x2
xy
-xy
-yZ
y
1, Multiply the last terms:
+ y(x - y)
+
= _y2
Add up the boxes:
Product:
=X2 _ y 2
+ yx ­ y 2 = x 2 - y 2
2
X - xy
x 2 +xy-xy-l=X2 _ y 2
Answer: x 2 - y 2
The product of the sum and difference of the same terms
always results in the difference of two squares. The
middle term is always zero.
Square of a Binomial:
(x
+ y)2
= x2 + 2xy + y2 and (x _
To find the first term, square the
first term of the binomial:
+ y)2 or
(x
(x)(x)
(x _ y)2
x
o Multiply the outer terms:
or
I Multiply the inner terms:
-2xy
To find the third term, square the
second term of the binomial:
= y2 or (_y)( _y) = y2
Products:
(x
+ y)2
xy
or
= yx or
(x)( -y)
(-y)(x)
=
=
= y2 or
(-y)(-y)
+
-xy
(x - y)2 = (x - y)(x - y)
2
=x -2xy+l
+ y)2
= x
=l
Add up the boxes:
+ xy + y 2 = x 2 + 2xy + y 2
X2 + xy
2
1. When you take the square of a binomial, the trinomial
product is known as a perfect square trinomial.
2. When squaring a binomial, the first and third terms are
always positive, and the middle term is twice the product
of the two terms involved.
226
Chapter 6: Operations with Polynomials
xy
-yx
+ xy + yx + l
2
=x +2xY+l
2
(x - y)2 = x - xy - yx + l
= x 2 _ 2xy + y2
(x
y
I--~t------!
y
1, Multiply the last terms:
(y)(y)
+
x
Products:
+ y)(x + y)
2
x + 2xy + l
= (x
=
(y)(x)
=
+ y2
Find the product: (x + y)(x + y)
= x2
To find the middle term, double
the product of the two terms of
the binomial:
(x)(y)
2xy
Box Method
E Multiply the first terms:
i
(y)(y)
= x2 -
FOIL Method
Distributive Method
2xy
y)2
PRACTICE
1. Which is the simplified form of
(x + 10)(x - 1O)?
2
B.
+ 20x + 100
x 2 + lOx + 100
C.
x2
D.
2
A.
X
x
100
lOx
-
+ 100
11. (2x + 7)(2x - 7)
12. (a + 2b?
13. (xy + 5z)(xy - 5z)
14. (2xy + 1)2
15. (3ab + 1)(3ab - 1)
2. Simplify (2m + 3)2.
A.
B.
C.
D.
+ 6m + 9
+ 12m + 9
+9
+ 12m + 9
2m 2
2m 2
4m 2
4m 2
3. Simplify
A.
(? - 2)(? - 2).
2
ru2 - 4v + 4
16.
(~ + ~ x) (~ - ~ x )
17. (2.1g - 3)(2.1g + 3)
18. (x 3 + 2)(x3
19. (2a -
2)
-
~)( 2a + ~)
20. 2x(3x + 1)2
21. (O.sx - 4y}2
4
8
4
4
B. ~-ru+4
c. gzi-ru+ 4
D.
+ --4J
4
4 2
--4J
9
3
­
4
4. Find the product of (1.2a + O.3b) and
(1.2a - 0.3b).
A.
B.
C.
D.
0.0144a 2 - 7.2ab + 0.09b2
1.2a2 + 0.3b 2
1.44a2 - 0.09b 2
1.44a2 + 7.2ab - 0.9b 2
5. Which of the following is a perfect square
trinomial?
2
+ 16x + 9
A.
x
B.
x2
C.
25x2
D.
9x
8x - 16
-
2
+9
+ 12x + 4
ElD Exercises 6-33: Multiply the expressions
and simplify the result. Show your work.
22.
(~ x -
23. (l -
2
Y
4x)2
24. (x 3 + 5y?
25. (2a - 3x3)(2a + 3x3 )
26. (5x 2 + 3y?
27. (2x - 7y3?
28. (x - a? + (a - x}2
29. x(x - 1)2 -
(x
+ 1?
30. (x + l)(x - 1)(x2 + 1)
31. (a - b)(a + b)(a 2
-
b2 )
+ 2)(a - 2)(a 2 + 4)
33. (x 2 + 9)(x + 3)(x - 3)
32. (a
34.
EIiIlIJ The pattern of the product of the
sum and difference can be used to multiply
certain number pairs. Use the pattern to find
the products of:
6. (mx + 2)(mx + 2)
a (13)(7)
7. (3x - 3)2
b (21)(19)
c (35)(25)
8. (l2c + 4)(l2c - 4)
d (27)(13)
9. (13x + 30)(13x - 30)
e (42)(38)
10. (O.4a - 0.02?
X
f (101)(99)
Special Products of Binomials
227
LESSON 6.6
6.6 Negative Integers as Exponents
If n is an integer and x =/= 0, then x- n
=
In
and ~ =
X
X n
xn •
x- n is the reciprocal of x n • This definition means we can now write any expressions
containing negative exponents without negative exponents. In other words, any
algebraic expression with a negative exponent can be written as the reciprocal
with a positive exponent.
To Simplify with Negative Exponents
• Subtract the exponent of the divisor from the exponent of the dividend,
3
such as x7 =
X3-
7
=
x- 4 .
X
• If the exponent is negative, take the reciprocal to rewrite the expression
with positive exponents, such as x- 4
=
~.
X
Model Problems 1-9: Perform each operation.
1. 37
•
3- 4
SOLUTION
37 +(-4) = 33
= 27
34 +(-4)
= 3° •
2. a- 2 • b- 4
3. 34
•
3- 4
•
3°
2
4. _1_ + (103 )-1
T
5. (-5)0 - 5-
2
•
1
32 + 103(-1)
=1 • 1=1
= 9 + 10- 3 = 9 + _1_3 = 9 + _1_
1
1-
1
"52 = 1 -
10
1
25
1
25 = 25 - 25
6. (a- 4 )-3 + (a 2)6
7. (X-3)2. (y6)-1
8.
x
-6
•y
1
1
1
- 6"" • 6"" or - (
)6
X
Y
xy
-6 _
(~r2(~)-5
9. (x - y)-2
228
Chapter 6: Operations with Polynomials
1
-,-----,--,,or
(x - y)2
1
2
x - 2xy
----=----""7
+l
24
= 25
1,000
PRACTICE
Exercises 1-15: Simplify. Write the variable
answers with positive exponents.
1. 5- 3
•
5s
Exercises 17-23: Find the value of x that makes
each statement true.
17. 3x
•
34 = 312
2. -3b- 3
3.. 7 • 10- 3
4 _9_
• 10- 2
5
22. MP 1,2,7,8
23. MP 1,2,7,8
8.
(~)-2 + (~y
9.
(~)-3(~)-2
10.
((X- 1)_2)-3
16. When x = 5, what is the value of
5x o - (5x)O
+ 5x- 1?
Negative Integers as Exponents 229
LESSON 6.7
6.7 Dividing Polynomials
Consider the Laws of Division for Exponents:
For all positive integers m and n, and for every nonzero real
am
n
number a,-an = am ­
.
x5
x
Sometimes the division results in a
positive exponent.
For example, 3 =
Sometimes the division results in a
negative exponent.
For example,
Sometimes the division results in both
positive and negative exponents.
For example,
Sometimes the division results in an
exponent equal to O.
For example, aO = 1,6° = 1, 100° = 1, (-7)° = 1,
(x + y)O = 1, and (4x 2y)0 = 1.
5 3
X -
= x2•
4
Y6
y
is y 4 - 6 =
y-2,
which can be written
8x 4m2 - 1X4 - 1 - x3
­ 2m 4 '
6 2m 6 - 2
16xm
• 0° is undefined.
Dividing a Monomial by a Monomial
The rules for division follow from the fact that division is the inverse operation of
multiplication.
To Divide a Monomial by Another Monomial
• Divide the numerical coefficients, using the law of signs for division.
• Divide the variable factors that have the same base using the laws of
exponents, subtracting the exponent in the denominator from the exponent
in the numerator.
• Simplify by multiplying the quotients that remain.
• Check the result by multiplying the divisor and the answer (or quotient).
If the answer is correct, then the product should produce the original
numerator.
230
Chapter 6: Operations with Polynomials
as~.
y
~ Model Problems 1-5: Simplify each expression. Show your work.
SOLUTION
24
XS
e = -
8
_
e X S 2 = -8x 3
-1
Check. (-3x 2)e(-8x3 ) = 24xs
= _
-3
=
8-8 e
x2
a3
a3
2
e
bb
= (-l)(l)b = -b
Check. (8a 3b)( -b) = -8a3b2
-3b
- 3x2-4bs-4 -- - 3x -2b or-­
x2
o
2
a(a a+ 1) - (a 2 + 1)0
~ (-2a 3 b2x4 ?
4
1
W
16
4
16a6b4x8
PRACTICE
EIiII
Exercises 1-30: Simplify. Express all
answers with positive exponents. Show your
work.
1.
2.
3.
y2k+l
10.
102x +1
11. lox+l
a16
a4
mS
12.~2
-2x
m
X
W
-2
x
yk
(where w > 2)
'JY
4. 7X (where x < y)
5. yX (where x> 1)
y
4a
6. 4b (where a > b)
13.
-0.08a3x 2y 4
0.2axy
2
a4b3c7
14. -3-4­
ac
15.
-18r4?
-3rc
16.
0.6a7x 2
0.2a 2x
am
17. - (when m > n)
an
a
7. a4
8.
9.
1
----,0,-----­
x +1
19.
-45x a + S
9x 3
Practice Problems continue . ..
Dividing Polynomials
231
Practice Problems continued . ..
20.
48a 2x
6x 3
4Sa 2zO
21. - - 4 ­
-Sa z
26.
aOb n
aXa
27.
4(x 2)3(y4)2
2(X2)2(yZ)4
22.
32 (xy)nJ z
8x nJ y nJ
28.
-9x3a
-3xa+ 3
23.
30xy 2
-lOx 2y3
29.
(-5 )(x2a )2y4b-l
2SX4a - 1y 4b
24.
-6a10m 4
-30a7 m S
30.
x 2(1 - x 2) _ (y2)0
x2
4xnJ
nJ
25. ~(whenm > k)
x Y
Dividing a Polynomial by a Monomial
In general, for all real numbers a and b, and all nonzero
a+b
a
b
a-b
a
b
real numbers c - - = - + - and - - = - - -.
'c
c
c
c
c
c
To Divide a Polynomial by a Monomial
• Divide each term of the polynomial by the monomial, using the distributive
property.
• Combine the quotients with correct signs.
• Check by multiplying.
Lastly, we need to consider the property of closure with respect to polynomials.
Recall that closure for integers means that when you add, subtract, or multiply two
integers, the result is always another integer. Polynomials are also closed under
addition, subtraction, or multiplication, since the sum, difference, or product
of polynomials is always another polynomial. In contrast, with both integers and
polynomials, the
di~ision operation is not closed. For instance, ~
yields a non­
integer result, and x9 results in a negative exponent, and polynomials must have
x
positive exponents.
232
Chapter 6: Operations with Polynomials
Model Problems 1-3: Simplify.
SOLUTION
2
6m= -
-m
-m
+ (-m)
--m
Check. -me -6m
3.
4a 2x - 8ax + 12ax 2
4ax
=
+ 1 = -6m + 1
-6m(2-1)
+ 1) = 6m 2
2
m
-
2
= 4a x _ 8ax + 12ax = a _ 2 + 3x
4ax
4ax
Check. 4ax(a - 2
4ax
+ 3x) = 4a 2x - 8ax + 12ax2
PRACTICE
2
1 S·
n 6x -6x
+ 12x
• Imp I y:
2
A.
1-2x
B.
-1 - 2x
C.
-x
D.
2x -1
4a 2
3
8.
+ 2x2
10.
242
33
A.
ax +ax
2
ax
aSx - 2a 2x + a3x 3
B.
a3x - 2a 2x 2
C.
a3 -
2a 2x
+ ax2
D.
a2
2a2 x
+ ax
+ ax 3
Exercises 3-20: Simplify the following expres­
sions by dividing by the monomial or constant.
3.
p
6.
7.
24x3 - 12x2 + 15x
12. ------'------'---....:....--.
3x
13.
14.
8x 3
14x2 + 2x
-2x
-
10a3b3
-x3
-
5a2 b2
5ab
+ 5x4
-
+ 15ab
6x5
-x3
Y
x +
x
16. -x-+­
x
Q
a
-
18.
8x3 - 6xy
18xy
6
8x6
4
2x
-
-
-
2x4
27a 2b2
-3ab2
+ 6a 3bs
3.4a 8b9clO - 5.1a6b2c8
19.
-1.7a 6b2c8
2x
-
6x 8
9a sb3
-a
12x2
'TTrh + 4'TTrh
'TTrh
-x
15.
p + prt
4a2
-1
5x 3 - 4x2 - 2x
11. - - - - ­
17.
5.
c2
-
2d - cd2
9.--­
cd
5
· l'fax
2 • SImp
I y:
-
+ 3b 2
+ 24y2
5a4 - 15a3 + 45a 2 - lOa
20. -----'------­
-5a
Dividing Polynomials
233
CHAPTER 6 REVIEW
1. If Dr. Beck has a weekly income of x dol­
lars and his average monthly expenses are
m dollars, which of the following algebraic
expressions represents the amount of money
he saves in one year?
A.
B.
C.
D.
12(4x - 12m)
52m - 12x
52x - 12m
12m - 12x
5. Which of the following equations is correct?
C.
(xaf. (x b ) = x 2ab
(x 2)3. (xaf = x6 + 2a
D.
(X n )3.
B.
(x 3)n = x9n
6. Simplify: 2a + 2a + 2a + 2a =
2. Evaluate each of the following.
a 4- 2
b 10-3
C 2 X 10- 2
d -10- 2
e
f
-S2
:)-3
h (-4)-3
7° + T 2 X 7
-1 4 X (-1)5
15x- 1
h
3x- 2
(3x- 4 f
(xy)4 -7- (xy? X (0.2)-3
4. Which expression is a perfect square
trinomial?
C. x 2 +1+y2
A. x 2 + X + 1
x 2 + 2x + 1
A.
6- 1
B.
63
C.
6- 2
•
64
D.
6- 3
•
6
•
•
6- 1
6- 5
8. In each of the following:
3. Simplify and express the following with
positive exponents.
a 52 X 57
b (S2)3
C
(10 X 103 )2
d x- 3
e 3a- 2
f a- 1b 2
4x 3
g y-2
B.
S4a
7. Which of the following expressions is not
1
equivalent to 36 ?
(-sf
g (_
D.
D.
x 2 - 4x - 4
(1) Write the polynomial in standard form.
(2) Name the coefficients of each term.
(3) State the degree of each term.
(4) State the degree of the polynomial.
a
3x - x 2 + 5x3 - 2 + 4x 5
b
3ab + 3ab 2 + ab 3 + b4 + a
Exercises 9-41: Simplify each expression.
9. tax - ax
10. 2x 2 - Sa 2 + 5x 2 - 4ax - 3a 2 + 4a 2 - 5ax
11. 3a - 4 - (-2a 2 - a)
12. 7x 2 - 3x + 5 - (7x 2 - 5x - 5)
13. (mr)(mr)(-m)
14. (- x )2(X 2y)( -y)3
15. -3(x - y)
16. x(a + b - 1)
17. 4x(x - 1)
Chapter Review continues . ..
234
Chapter 6: Operations with Polynomials
Chapter Review continued . ..
43. ~ Molina's friend, Thor, said
that 62 - 52 is 36 - 25 or 11, and that 6 + 5
is also 11. He noticed that a similar result
occurred for 9 and 10: that
102 - 92 = 100 - 81 or 19, and that
10 + 9 = 19. So, using those examples
and many others, Thor concluded that the
positive difference between the squares of
two consecutive integers is always equal to
the sum of those two integers. Show that,
algebraically, he's right.
18. X 3 (X 4y - 22)
19. 3a 2(b - k - 1)
20. O.5ab( -8a + 2b + 5)
21. -rrr2(h - 4r + 1)
22. 10"(10b + lOC)
23. (-3x 2y S?
24. (-2a 2 b 2 )3 + (4a 3b3 ?
25. (x + 2)2
44.
26. (x - 2y?
lOa + 8 and the length is 4a + 7.
27. (3x + 7)2
a What is the algebraic expression for the
width of the rectangle? Show your work.
28. (3x - 7)(3x + 7)
b What is the area of the rectangle in
simplest form?
29. 3x - 7(3x + 7)
30. (2n + 3)(n 2
4n - 1)
-
3 2
36a b
31. -9a 2b
7a 3b 2cS
32. -28a 2b2c2
10x3a
33.
2xa
2x 2+ ­ 4x
34. x
36.
-
3a 3
45.
mZJEJ
46.
EIiIJ The height in feet of a ball tossed
If n is an even number and n + 1
is an odd number, show algebraically that
the square of an odd number is also an odd
number. State your argument clearly.
in the air is given by the function
h(t) = -16f + 32t + 4, where t is time in
seconds since the ball was tossed.
a Find the height of the ball 1 second after
it was tossed.
-rrr - 2-rrr
35. - - - ­
2a 4
EIiIJ The perimeter of a rectangle is
-
b Find the height after 0.5 seconds and
1.5 seconds.
4a 2
2
-a
c What conclusion can you draw from the
answer to part b?
x 2 + 7x - 8
x+8
2
a - Sa - 36
38. - - - - ­
a+4
37.
47.~
40. a - (a - 5)(a + 2)
a Use your arithmetic understanding
in adding two fractions with different
denominators and write a single fraction
that expresses X-I + y-I.
41. 2[x2 + 3(x - 2)] - 3[x2 + 3(x - 1)]
b Give a counterexample to show why
39. 6b + 2b( -3b + 2) + 5b
2
42. ~ Molina noticed that when she
had three consecutive integers, such as 3, 4, 5,
that (3)(5) = 15 and 42 = 16. And, again, with
6, 7, 8, that (6)(8) = 48, while 72 = 49. After
working with both positive and negative
integers, Molina offered the following con­
jecture: Given any three consecutive integers,
starting with x, the product of the first and
the third is one less than the square of the
second. Clearly identify your own variables
and show that, algebraically, she's right.
X-I
+ y-I
1= _1_. Use your result in
x+y
part a as another way to support your
answer.
Chapter 6 Review
235