Download b. (If=(y*y)(y*y)(y*y)(y*y) =y*y*y*y*y*y*y*y = yB =x5/ 2. a. (2xi)6

Review Chapter 6
1. a. X4*X5=(x*x*x*x)(x*x*x*x*x)
=x*x*x*x*x*x*x*x*x
=x9
b. (If=(y*y)(y*y)(y*y)(y*y)
=y*y*y*y*y*y*y*y
= yB
c. (x *y)5 = (x * y)(x * y)(x * y)(x * y)(x *y)
=x*y*x*y*x*y*x*y*x*y
= (x *x* x* x* x)(y* y* y* y* y)
=x5/
2. a. (2xi)6 =26x6vt
=64x6i8
b. 4y*7/
=4*7* y* l
=28i+5
=28l
3. Volume = side3
= (2.1 * 104 inches
Y
= 2.13 *(104)3 inches3
= 2.13 * 1012 inches3
= 9.261
* 1012 inches3
98
Chapter 6: Exponents and Factoring
.
s
4. a. X4X-7 = X4+(7)
=X"3
8.
a.
2
xy
4
--x S-2y 1-4
=X3y"3
=( 3)C3)
b.
X -4X7
=X -4+7
=-X3
l
=X3
Y J4x2Y
y3
b. (4x2J
=43(X2)3
l
64x6
-7
5. distance = rate*time
;;
(
C. --,-- x 3-7y 2-lf
_(
XY
xy
3 2J
1.1341103feet/second* 10-6seconds
=(X"4/f
=1.13*1034110-6 feet
'" x -4'4i'4
s 1.13* lOJ+n) feet
= X"16
y4
= 1.13* 10 ) fool = .0013 loot
x7
6. a. -r;;;x
x
7-3
=C6
4
""x
)(
4
)
=L
XI6
c.
(;J = ;:
7. a. 3. Quotient law
b. 5. Power-of-a-power law
c. 2. Quotient law and defmition of negative
integer exponents
d. 1. Productlaw
e. 4. Negative exponent in the denominator
_27l
-
8XI2
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--
- .--
99
Chapter Review
9.
5...
13. The GCF of5, 15, and 10 is 5.
The powers of x are ~, ~, xS. The smallest power
is~.
Zx'y
The powers of y are y1,l,
4~1 wx' G
4xJ(5x + 2x4y) = 20x4 + 8x7Y
10. a. 5x(6x- 1) = 30x2- 5x
=~z
b. 2xz(4x+ 7y-3)
+ 14xyz- 6xz
c. -(x2-8x+2) =-~+8x-2
11. a. The powers of x are Xl and xS. The smallest
power is x.
The powers of y are l and yl. The smallest
power is y.
Taking the smallest powers of x and y gives a
GCF of x*y
= xy.
b. The GCF of 18,3, and 9 is 3.
The powers of x are ~, x3,x4. The smallest
power is~'
y and z are not factors of all three expressions,
so they don't appear in the GCF.
Taking the;smallest powers gives a OCF of
3"'~ = 3x1.
c. Using the factor tree we have 60 = 2~*3*5and
105 = 3*5*7.
2 is not a fRctorof IOSand 7 is not Iifactor of
60, so 2 and 7 do not appear in the GCF.
The powers oD arc 31 and 31. The smallest
power is 3.
The powers of5 are 51 and 51. The smallest
poweris 5.
x andy are not factors of both expressions, so
they don't appear in the GCF.
Taking the smallest powers gives a GCF of
3.5 = 15.
60
-
5~#
The smallest power
3xy3 + ~y)
14. a. -2x(4x- 5)
b. -3(3x+ 4y)
15. a. Thetwotermshavea GCFofz. Factoring
gives:
z(4x - 3)
b. We have two terms 4x(y + 7) and -3(y + 7).
Common to both terms is the binomial
(y + 7). Factoring (y + 7) from both terms
gives (y + 7)(4x -3).
c. (a) and (b) are basically the same. Let z
replace y + 7 in part (b).
16. a.
~
x2 + 5x .. 0
x(x + 5) = 0
x;;;Oorx+5=O
x""Oorx"'-S
~-8x=0
~x-~=O
x=Omx-8=0
x=Omx=8
17. a.
...
2
...~
A 30
,~
2
21'.15
(x+4)(x+
31'.5
;.
isy.
z is not a factor of all three expressions, so it does
not appear in the GCF.
Taking the smallest powers gives a GCF of
5*~*y = 5~y.
Factoring the GCF from all three terms, we have
2) = x2 + 6x+ 8
b.
5...
12. The GCFof 8; and 12yis 4y.
...
US
sx:
..
-6x
;
"I
:~
I~i!
8y2 + 12y = 4y(2y
+ 3)
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'"'12.tt
.'
-15:.-
18
(x-3)(4x2 + 5x- 6) =4xJ -7x2 -21x+ 18
100
Chapter 6: Exponents and Factoring
18. a. By the diagonal test for grouping, any
placement of tenns in the area diagram that
gives equal diagonal products will work. One
arrangement has xy and "18along one diagonal
and 2x and "9y along the other diagonal.
x
-9
19. a. Factorable; the product of the fIrst term and
the last term 4x3*"5= "20x3,equals the product
of the middle two terms
IO~*-2x = -20~. Thereis no needto
rearrange these terms because the product of
the fIrst and last term equals the product of the
middle two terms.
4x3 + 10x2- 2x- 5
=(4x3 + 10x2) + C2x2
2x
=2x2(2x+5)+ -1(2x+5)
2x - 9y + xy - 18 = (y + 2)(x - 9)
To factor symbolically, we want the product
of the fIrst and last terms to equal the product
of the middle two terms. Several groupings
will accomplish this.
2x-9y+xy-18
=xy-9y+2x-18
= (xy-9y)+ (2x-18)
= y(x-9)+ 2(x -9)
= (x - 9)(y + 2) or (y + 2)(x - 9)
b.
:J,r
-1)'2
"';'tV
3.l
I
- 4y + 3z -12yz
b. Not factorable; we can't pair terms to make the
product of one pair equal the product of the
other pair.
c. Factorable; the product of the fIrst term and
the second term 3x*8y = 24xy, equals the
product of the third term and the last term
6xy*4 = 24xy.
3x + 8y + 6xy + 4
Arranging the terms so that the fIrst and last
term have the same product as the middle two
terms. we have
6xy + 3x + 8y + 4
= (6xy + 3x) + (Sy + 4)
= 3x(2y + I) + 4(2y + 1)
(2y + 1)(3x + 4)
20. a. (x + y)2 = x2+ 2xy + l
+ 1 =C 4y + 1)(3z + I) Factorin
b. (X+2y)2 =x2 +2*x*2y+(2y)2
=x2 +4xy+4y2
c. (5x-3y)2 =(5x+ -3y)2
= (5X)2+2*5x* -3Y+C3y)2
=-12yz-4y+3z+1
+ (3z + 1)
= -4y(3z+I)+1(3z+1)
=(3z + I)C4y+ 1) or C4y+
= (2x + 5)(2x2 -1)
·
)
i'
g symbolically gives
-4y+3z-12yz+l
=C12yz - 4y)
5)
=25x2 -30xy+9y2
1)(3z+ 1)
21. a. The diagonal product is the product of the fIrst
and last term: ~*24 = 24x2.
b. Ix and 24x, 2x and 12x, 3x and 8x, 4x and 6x
c. 3x and 8x sum to Ilx.
d. x2 + llx + 24 = = ~ + 3x + 8x + 24
e.
x
8
r
xI
3~
24
+ llx+ 24 = (x+ 3)(x+ 8)
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--
---
-=
-
Chapter Review
23. a. Diagonal product = ~*5 = 5~
The diagonal product is positive and the
middle term is negative, so both factors must
be negative.
'I * '5 = 5 'Ix + -5x = '6x No
There are no factor pairs of 5~ that add to
equal the middle term, '2x, so ~ - 2x + 5 does
not factor. This is called a prime polynomial.
22. a. Split the middle term by fmding a factor pair
of the diagonal product, -14~, that adds to
equal the middle term, -5x. The diagonal
product is negative so we need one positive
factor and one negative factor. The middle
term is negative so the negative factor must
have a larger absolute value than the positive
factor.
1*-14 = '14
Ix + '14x = '13x No
2x + -7x = -5x Yes
2*-7 = '14
x2-5x-14=
~+2x-7x-14
= x(x + 2) - 7(x + 2)
= (x + 2)(x - 7)
b. Diagonal product = ~*6 = 1~
The diagonal product is positive and the
middle term is positive, so the both factors
must be positive.
1 * 12
b. Diagonal product = 5x2*-2 = -1O~. Look for a
factor pair of 1O~ that adds to the middle
term, 9x. The diagonal product is negative, so
we needonepositivefactorandonenegative
factor. The middle term is positive so the
positive factor must have a larger absolute
value than the negative factor.
-1*10
= -10
-2*5= '10
5~ + 9x
'Ix + lOx
= 9x
Yes
-2x + 5x = 3x No
- 2 = 5x2 - Ix + lOx - 2
"'x(5x-l)+2(5x-l)
=(5x - 1)(x + 2)
d. Arranging the terms in descending order
7~ + 26x - 8. The diagonal product
= 7x2*-8= -56x2. Look for a factor pair of
-56~ that adds to 26x. The diagonal product
is negative, so one factor is positive and the
other is negative. The middle term is positive,
so the positive factor must have a larger
absolute value than the negative factor.
-1*56 = -56
-Ix + 56x = 55x No
-2*28 = -56
'2x + 28x = 26x Yes
-4*14 = -56
'4x + 14x = lOx No
-7*8 = -56
-7x + 8x = Ix No
7x2 + 26x
- 8 = 7x2 -
.
= 12
2x + 28x - 8
=x(7x - 2) + 4(7x - 2)
=(7x - 2)(x + 4)
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Ix + 12x
= 13x
no
2 * 6 = 12
2x + 6x = 8x no
3 * 4 = 12
3x + 4x = 7x no
There are no factor pairs of 1~ that add to
equal the middle term, x, so ~ + x + 6 is a
prime polynomial.
24. a. Diagonal product = ~*21i = 21x2i
Look for factor pairs of 21x2i that add to
lOX)'. The diagonal product and middle term
are positive so both factors arc:positive.
J*21.. 21
lxy + 21xy= 2~' No
3*7'" 21
c. Diagonal product - 3~*' 4 = l2x2. Look for a
factor pair of -12x2that adds to 11x. The
diagonal product is negative so we need a
negative and positive factor. The middle term
is negative so the negative factor must have a
larger absolute value than the positive factor.
1*-12=-12
lx+-l2x='llx
Yes
2*-6='12
2x+-6x=Ax
No
3*A = '12
3x + Ax = -Ix No
3x2- llx - 4 = 3~ + Ix - 12x - 4
=x(3x+ 1)-4(3x+ 1)
=(3x+ 1)(x-4)
101
.
3xy+ 7,ty= lO.\;yYes
~+ lOxy+2ly
...~ + 3xy + txy + 21l
= x(x + 3y) 7y(x + 3y)
= (x + 3y)(x + 7y)
b. Diagonalproduct= 2x4*-lS'" -30x4
Look for factor pairs of'30x4 that add to "x'-.
The diagonal product is negative, so one factor
is positive and the other negative. The middle
term is negative so the negative factor must
have a larger absolute value than the positive
factor.
1*-30 = -30
l~ + -30~ = -29x2 No
2*-15
= -30
2x2 + -15~ = -13~
No
3*"10= -30
3x2+ -10~ = -7~ No
5*-6= '30
5~ + -6~ = -lx2 Yes
2x4-~ -15 = 2x4+ 5~ - 6x2- 15
=x2(~ + 5)- 3(~ + 5)
= (~ + 5)(~ - 3)
25. a. ~ is the squareofx and25 is the squareof 5.
x2- 25 = (x - 5)(x+ 5)
b. 9 is the squareof 3 andi is the squareofy.
9 - i = (3 - y)(3 + y)
c. 4~ is the squareof 2x and81i is the square
of9y.
4~ - 81i = (2x- 9y)(2x+ 9y)
102
Chapter 6: Exponents and Factoring
- 36x + 16
=2(9x2 - 18x + 8)
=2(9x2 - 6x - 12x + 8)
=2(3x(3x - 2) - 4(3x - 2))
=2(3x - 2)(3x - 4)
29. a. 6 - 48z3 =6(1-
b.
8Z3)
=6(1-
26. a. 182
-
2z)(1 + 2z + 4Z2)
= 2(125x3+ i)
b. 250x3 + 21
=2(5x + y)(2Sx2 - Sxy+ l)
c. 64-/
=(8-l)(8 +i)
=(2- y)(4+2y+l)(2+ y)(4-2y+ l)
x/- xy + 2/- 2y
=y(xy - x + 2y - 2)
= y(x(y - 1) + 2(y - 1))
= y(y - l)(x + 2)
Chapter 6 Test
c. 32y - 300y
= 3y(x2 - 100)
= 3y(x - 10)(x + 10)
1. (xy3Z-4f=x2(l)2(z-4f
27. a. Attempt to factor the left side of the equation.
The diagonal product is 40x2. Look for a
factor pair of 40x2 that adds to 13x. Both the
diagonal product and the middle term are
positive, so the both factors are positive. Of
the four factor pairs of 40x2,only 5x and 8x
add to 13x.
x2 + 13x + 40
x2 + 5x + 8x + 40
=0
x(x + 5) + 8(x + 5)
=0
=X2y6z-g
x2/
=7
2.~
6 -3
20x-2
3x3y-3
=0
=10
3x3
= 10y3
(x+5)(x+8)=0
x+5=Oorx+8=0
x=~50rx=~8
b.
"
x2 -+-2x '" 3
x2+2x-3"'O
"Ix and 3x is the factor pair 0f -3x2that adds to
2x.
x2 - Ix + 3x - 3 = 0
x(x-1)+3(x-1)=0
(x-1)(x+3)=0
x-1=00rx3=0
x = lor x = -3
3.
:" -,
l3:)-)
- =(3X4~lyHf
-3
=(3xJl)
-3
= (3x3 * 1)
-3
=(3i)
=3 -3(x
3 -3
)
c.3x2-19x+6=0
"Ix and "18x is the factor pair of 182 that adds
to "19x.
3x2 -Ix -18x + 6
x(3x -1) - 6(3x -1)
(3x -l)(x - 6)
3x-1=00rx-6=0
3x = 1 or x = 6
x=-
=0
=0
=0
1
orx=6
3
28. a. x3+ 64 = (x + 4)(2 - 4x + 16)Sumof two
cubes a = x, b = 4
b. 1 - /
= (1 - y)(l
+ Y + l) Difference of two
cubes a = 1, b = y
c. 8x3+ 1000/ = (2x+ 10y)(4x2- 20xy + 100l)
Sum of two cubes a = 2x, b = 10y
4.
signals
= 2.0 * 107 signals/second * 8.64 * 104 seconds
=2.0*8.64*107
=17.28*107+4
*104 signals
signals
= 17.28 * 1011signals
= 1.728
* 1012 signals
5. a. 3a*(a + 4) = 3a2 + 12a
b. -6xy2 * CX2 - 2xy + Sl)
=6x3l
+ 12x21- 30xl
d. / + 27i = (y + 31)(y2 - 3yl + 9i) Sumof
two cubes
a
= y, b = 31
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Chapter Test
6. -6ab2c3 -3bc2 _12a3b4c3
= -3bc2(2abc+1 + 4a3b3c)
7. 2x2-10x=0
2x(x- 5)= 0
2x=0 or x-5=0
x=O or x=5
8. a. (2x + 5)(x-7) = 2x(x -7) + 5(x - 7)
=2x2-14x + 5x-35
=2X2-9x-35
b. (x-4)(3x2 -x+6)
=x(3x2 -x+ 6) - 4(3x2 - x + 6)
=3x3 - x2 + 6x -12x2 + 4x - 24
=3x'
_13x2 + 10x- 24
9. a. 2x+ 4xy + 3... 6y = 2x(1+ 2)') + 3(1+ 7y)
. (1+ 2y)(2x + 3)
b.
6x' - 9x - 2X2 or3 = 6x3 - 2X2 - 9x + 3
'" 2x2(3x -1) - 3(3x -I)
'" (3x - 1)(2x2 - 3)
10.
(2x-7y)2
",(2x+ -7y)2
a (2X)2 +2*2x.C7)')+C7y)2
'" 4x2
-
28xy + 49y2
11. a. 4x2-23x-6=4x2 +1x-24x-6
=x(4x+I)-6(4x+1)
= (4x+
b. x2+ 2xy -351
1)(x-6)
=x2 -5xy+
=x(x-5y)+
7xy- 35y2
7y(x-
5)')
=(x-5y)(x+7y)
12. 64x2-l
=(8x - y)(8x + y)
13. a. 2x2y+6xy2-80l
=2y(x2 +3xy-401)
=2y(x2- 5xy+ 8xy- 401)
=2y(x(x -5y) + 8y(x-5y))
=2y(x- 5y)(x+ 8y)
b. 12-3x2 =3(4-X2)
=3(2-x)(2+x)
6x2-17 x + 5 =0
14.
6x2 - 2x -15x + 5 = 0
2x(3x -1) - 5(3x -1)
=0
(3x-1)(2x-5)=0
3x -1 = 0 or 2x - 5 = 0
3x = 1or 2x = 5
,
1
5
x=- orx=3
2
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103