Chapter 7 52. 48
47. a. ( x + y )( 2x − y ) = 0
x+y=0 x+y−y=0−y
1( 48 ) = 48 2( 24 ) = 48
2x − y = 0
or
2x − y + y = 0 + y
x = −y
2x = y 2x 2
y 2 1 x = —y 2 1 So, the roots of the equation occur when x = −y and x = —y. 2 —=—
b. ( x2 − y2 )( 4x + 16y ) = 0
x2 − y2 = 0
4x + 16y = 0
or
x2 − y2 + y2 = 0 + y2
4x − 4x + 16y = 0 − 4x
x2 = y2 —
16y = −4x 16y −4x −4 −4 −4y = x
—
± √ x2 = ± √ y2
—= —
x=±y
So, the roots of the equation occur when x = ± y and x = −4y. 48.
( 4x−5
− 16 )
( 3x
− 81 ) = 0
4x−5 − 16 = 0
or
+16 +16
3x − 81 = 0 +81
+81
4x−5 = 16
3x = 81
4x−5 = 42
3 x = 34
x−5=2
x=4
+5 +5 x=7 The solutions are x = 7 and x = 4.
3( 16 ) = 48 4( 12 ) = 48 6( 8 ) = 48 So, the factor pairs of 48 are 1 and 48, 2 and 24, 3 and 16, 4 and 12, and 6 and 8. 7.1–7.4 What Did You Learn? (p. 383) 1. Sample answer: Use x + 4 to represent each side of the
square. So, an expression that represents the area is ( x + 4 )2. Then use the square of a binomial pattern to simplify this expression into a polynomial. As another method, you could calculate the area of each shaded region and add those areas together. 2. Sample answer: The solutions are the constant terms with
the opposite sign. This method does not work when the coefficient is not 1. 7.1–7.4 Quiz (p. 384) 1. The polynomial −8q3 is in standard form.
The only term has a degree of 3. So, the degree of the polynomial is 3. The leading coefficient is −8. The polynomial has 1 term. So, it is a monomial. 2. You can write the polynomial 9 + d 2 − 3d in standard form
as d 2 − 3d + 9.
The greatest degree is 2. So, the degree of the polynomial is 2. Maintaining Mathematical Proficiency 49. 10
1( 10 ) = 10 2( 5 ) = 10 So, the factor pairs of 10 are 1 and 10, and 2 and 5. 50. 18
1( 18 ) = 18 2( 9 ) = 18 3( 6 ) = 18 So, the factor pairs of 18 are 1 and 18, 2 and 9, and 3 and 6. 51. 30
1( 30 ) = 30 2( 15 ) = 30 3( 10 ) = 30
The leading coefficient is 1. The polynomial has 3 terms. So, it is a trinomial. 2 3
5 m6 + 2 m4. as − — — 6 3 The greatest degree is 6. So, the degree of the polynomial is 6. 5 The leading coefficient is −—. 6 The polynomial has 2 terms. So, it is a binomial.
4. You can write the polynomial −1.3z + 3z4 + 7.4z2 in
standard form as 3z4 + 7.4z2 − 1.3z.
The greatest degree is 4. So, the degree of the polynomial is 4. The leading coefficient is 3. The polynomial has 3 terms. So, it is a trinomial. 5. ( 2x2 + 5 ) + ( −x2 + 4 ) = 2x2 − x2 + 5 + 4
= ( 2x2 − x2 ) + ( 5 + 4 )
5( 6 ) = 30 So, the factor pairs of 30 are 1 and 30, 2 and 15, 3 and 10, and 5 and 6.
Copyright © Big Ideas Learning, LLC All rights reserved.
5 6
3. You can write the polynomial — m4 − — m6 in standard form
= x2 + 9 The sum is
x2
+ 9.
Algebra 1 Worked-Out Solutions
403
Chapter 7 6. ( −3n2 + n ) − ( 2c2 − 7 ) = −3n2 + n − 2n2 + 7
The difference is
−5n2
5x(x − 3) = 0
= −5n2 + n + 7
5x = 0 5x 0 —=— 5 5 x=0
+ n + 7.
)−( ( = −p2 + 4p − p2 + 3p − 15 −p2 + 4p
7.
p2 − 3p + 15
)
3=0
8−g=0
The difference is −2p2 + 7p − 15. 8. ( a2 − 3ab + b2 ) + ( −a2 + ab + b2 )
= a2 − a2 − 3ab + ab + b2 + b2 = ( a2 − a2 ) + ( −3ab + ab ) + ( b2 + b2 ) = 0 − 2ab + 2b2
+g +g
+g +g
8=g
8=g
The equation has repeated roots of g = 8. 17. (3p + 7)(3p − 7)(p + 8) = 0
−7
The sum is −2ab + 2b2. 9. ( w + 6 )( w + 7 ) = w( w + 7 ) + 6( w + 7 )
= w( w ) + w( 7 ) + 6( w ) + 6( 7 ) = w2 + 7w + 6w + 42 = w2 + 13w + 42 + 13w + 42.
8−g=0
or
3p + 7 =
= −2ab + 2b2
0
or
3p − 7 =
−7
+7
0
or
+7
O
I
L
= 3( 2d ) + 3( −5 ) + ( −4d )( 2d ) + ( −4d )( −5 ) = 6d − 15 − 8d 2 + 20d = −8d 2 + ( 6d + 20d ) − 15
−8
3p = −7 7 3p −— = −— 3 3 7 p = −— 3
3p = 7 3p 7 —=— 3 3 7 p=— 3 7 7 The roots are p = −—, p = —, and p = −8. 3 3 −3y = 0 or y − 8 = 0 or 0 −3y —=— +8 +8 −3 −3 y=0 y=8
The product is −8d 2 + 26d − 15. y2 + 2y − 3 y+9
×
—— 2 + 18y − 27 9y y3 + 2y2 − 3y
—— 3 + 11y2 + 15y − 27 y—— The product is y3 + 11y2 + 15y − 27. 12. (3z − 5)(3z + 5) = (3z)2 − 52
= 9z2 − 25 The product is 9z2 − 25. 13. (t + 5)2 = t 2 + 2(t)(5) + 52
=t 2
+ 10t + 25
The product is t 2 + 10t + 25. 14. (2q − 6)2 = (2q)2 − 2(2q)(6) + 62
= 4q2 − 24q + 36 The product is 4q2 − 24q + 36.
404
0 −8
p = −8
2y + 1 = 0 −1
= −8d 2 + 26d − 15
11.
p+8=
18. −3y(y − 8)(2y + 1) = 0
10. ( 3 − 4d )( 2d − 5 ) F
+3 +3
16. (8 − g)(8 − g) = 0
= −2p2 + 7p − 15
The product is
x−3=0
or
The roots are x = 0 and x = 3.
= ( −p2 − p2 ) + ( 4p + 3p ) − 15
w2
15. 5x2 − 15x = 0
= ( −3n2 − 2n2 ) + n + 7
Algebra 1 Worked-Out Solutions
1 The roots are y = 0, y = 8, and y = −—. 2
−1
2y = −1 2y −1 —=— 2 2 1 y = −— 2
19. a. P = 2ℓ + 2w
= 2(x + 72 + x) + 2(x + 48 + x) = 2(2x + 72) + 2(2x + 48) = 2(2x) + 2(72) + 2(2x) + 2(48) = 4x + 144 + 4x + 96 = 8x + 240 A polynomial that represents the perimeter of the blanket including the fringe is (8x + 240) inches. b. A = ℓw
= (x + 72 + x)(x + 48 + x) = (2x + 72)(2x + 48) F
O
I
L
= 2x(2x) + 2x(48) + 72(2x) + 72(48) = 4x2 + 96x + 144x + 3456 = 4x2 + 240x + 3456 A polynomial that represents the area of the blanket including the fringe is (4x2 + 240x + 3456) square inches. Copyright © Big Ideas Learning, LLC All rights reserved.
Chapter 7 c. P = 8x + 240
Section 7.5
= 8(4) + 240
7.5 Explorations (p. 385)
= 32 + 240
1. a.
= 272 A = 4x2 + 240x + 3456 = 4(4)2 + 240(4) + 3456 = 4(16) + 960 + 3456
Area = x 2 − 3x + 2 = (x − 1)(x − 2)
= 64 + 960 + 3456 b.
= 4480 When the width of the fringe is 4 inches, the perimeter of the blanket is 272 inches and the area is 4480 square inches. 20. a. 1000(1 + r)2 = 1000(1 + r)(1 + r)
F
O
I
Area = x 2 + 5x + 4 = (x + 1)(x + 4)
L
= 1000[ 1(1) + 1(r) + r(1) + r(r) ]
c.
= 1000(1 + r + r + r2) = 1000(r2 + 2r + 1) = 1000(r2) + 1000(2r) + 1000(1) = 1000r2 + 2000r + 1000 The polynomial in standard form that represents the balance of your bank account after 2 years is (1000r2 + 2000r + 1000) dollars. b.
1000r2 + 2000r + 1000 = 1000(0.03)2 + 2000(0.03) + 1000 = 1000(0.0009) + 60 + 1000
Area = x 2 − 7x + 12 = (x − 4)(x − 3) d.
= 0.9 + 60 + 1000 = 1060.9 When the interest rate is 3%, the balance after 2 years is $1060.90. c. 1000(1 + r3) = 1000(1 + 0.03)3
= 1000(1.03)3
Area = x 2 + 7x + 12 = (x + 4)(x + 3)
= 1000(1.092727)
2. Arrange algebra tiles that model the trinomial into a
= 1092.73 The balance after 3 years is only $1092.73. So, you do not have enough money for the $1100 guitar. 5
y = −— 216 (x − 72)(x + 72)
21.
0= 216
5 −— 216 (x 216
(
3. Find two integer factors of c that have a sum of b, then write
the binomial factors by adding each integer factor to x.
− 72)(x + 72) 5
)
7.5 Monitoring Progress (pp. 386–388)
— — −— 50 (0) = − 5 − 216 (x − 72)(x + 72)
1. x 2 + 7x + 6
0 = (x − 72)(x + 72) x − 72 = + 72
0 + 72
or
x + 72 = − 72
x = 72
0 − 72
x = −72
So, the width of the bunker at ground level is ∣ −72 − 72 ∣ = 144 inches.
Copyright © Big Ideas Learning, LLC All rights reserved.
rectangular array, use additional algebra tiles to model the dimensions of the rectangle, then write the polynomial in factored form using the dimensions of the rectangle.
Factors of 6
Sum of factors
1, 6
7
2, 3
5
So, x 2 + 7x + 6 = (x + 1)(x + 6).
Algebra 1 Worked-Out Solutions
405
