NAME ______________________________________________ DATE
5-1
____________ PERIOD _____
Skills Practice
Monomials
Simplify. Assume that no variable equals 0.
2. c5 ? c2 ? c2 c 9
1
a
3. a24 ? a23 }
7
4. x5 ? x24 ? x x 2
5. (g4)2 g 8
6. (3u)3 27u 3
7. (2x)4 x 4
8. 25(2z)3 240z 3
9. 2(23d)4 281d 4
11. (2r7)3 2r 21
k9
k
1
k
10. (22t2)3 28t 6
s15
s
3
12. }
12 s
13. }
10 }
14. (23f 3g)3 227f 9g 3
15. (2x)2(4y)2 64x 2y 2
16. 22gh( g3h5) 22g 4h 6
17. 10x2y3(10xy8) 100x 3y11
18. }
3 5 }
2
26a4bc8
36a b c
c7
6a b
19. }}
2}
3
7 2
Lesson 5-1
1. b4 ? b3 b 7
24wz7 8z 2
3w z
w
2
210pq4r 2q
25p q r p
20. }}
}
3 2
2
Express each number in scientific notation.
21. 53,000 5.3 3 104
22. 0.000248 2.48 3 1024
23. 410,100,000 4.101 3 108
24. 0.00000805 8.05 3 1026
Evaluate. Express the result in scientific notation.
25. (4 3 103)(1.6 3 1026) 6.4 3 1023
©
Glencoe/McGraw-Hill
9.6 3 107
1.5 3 10
10
26. }}
23 6.4 3 10
241
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-1
Practice
____________ PERIOD _____
(Average)
Monomials
Simplify. Assume that no variable equals 0.
1. n5 ? n2 n7
2. y7 ? y3 ? y2 y12
3. t9 ? t28 t
4. x24 ? x24 ? x4 }
4
1
x
8c9
6. (22b22c3)3 2 }
b6
5. (2f 4)6 64f 24
20d 3t 2
v
7. (4d 2t5v24)(25dt23v21) 2 }
5
4m 7y 2
3
3
7
227x (2x ) 27x 6
11. }}
}
16x4
16
12m8 y6
29my
10. }
7 2}
4
1 3r 2s z 2
12. }
2 3 6
256
wz
1
21
4
4
3
2
2 } d 5f
3
2
4
}
9r 4s 6z 12
14. (m4n6)4(m3n2p5)6 m 34n 36p 30
13. 2(4w23z25)(8w)2 2 }
5
3
2
s4
3x
26s5x3
18sx
9. }
4 2}
15. } d 2f 4
8. 8u(2z)3 64uz 3
1
212d 23f 19
(3x22y3)(5xy28) 15x11
(x ) y
y
6
2x3y2 22 y
}
2x y
4x 2
16. }
2 5
2
220(m2v)(2v)3
5(2v) (2m )
4v2
m
18. }}
2}
2
4
2
17. }}
}
23 4 22
3
Express each number in scientific notation.
19. 896,000
8.96 3 105
21. 433.7 3 108
20. 0.000056
5.6 3 1025
4.337 3 1010
Evaluate. Express the result in scientific notation.
22. (4.8 3 102)(6.9 3 104)
3.312 3 107
23. (3.7 3 109)(8.7 3 102)
3.219 3 1012
2.7 3 106
9 3 10
3 3 1025
24. }}
10
25. COMPUTING The term bit, short for binary digit, was first used in 1946 by John Tukey.
A single bit holds a zero or a one. Some computers use 32-bit numbers, or strings of
32 consecutive bits, to identify each address in their memories. Each 32-bit number
corresponds to a number in our base-ten system. The largest 32-bit number is nearly
4,295,000,000. Write this number in scientific notation. 4.295 3 109
26. LIGHT When light passes through water, its velocity is reduced by 25%. If the speed of
light in a vacuum is 1.86 3 105 miles per second, at what velocity does it travel through
water? Write your answer in scientific notation. 1.395 3 105 mi/s
27. TREES Deciduous and coniferous trees are hard to distinguish in a black-and-white
photo. But because deciduous trees reflect infrared energy better than coniferous trees,
the two types of trees are more distinguishable in an infrared photo. If an infrared
wavelength measures about 8 3 1027 meters and a blue wavelength measures about
4.5 3 1027 meters, about how many times longer is the infrared wavelength than the
blue wavelength? about 1.8 times
©
Glencoe/McGraw-Hill
242
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Skills Practice
Polynomials
Determine whether each expression is a polynomial. If it is a polynomial, state the
degree of the polynomial.
1. x2 1 2x 1 2 yes; 2
b2c
d
1
2
2. }
4 no
3. 8xz 1 } y yes; 2
Simplify.
5. (5d 1 5) 2 (d 1 1)
3g 1 12
6. (x2 2 3x 2 3) 1 (2x2 1 7x 2 2)
4d 1 4
7. (22f 2 2 3f 2 5) 1 (22f 2 2 3f 1 8)
3x 2 1 4x 2 5
8. (4r2 2 6r 1 2) 2 (2r2 1 3r 1 5)
24f 2 2 6f 1 3
9. (2x2 2 3xy) 2 (3x2 2 6xy 2 4y2)
5r 2 2 9r 2 3
10. (5t 2 7) 1 (2t2 1 3t 1 12)
2x 2 1 3xy 1 4y 2
11. (u 2 4) 2 (6 1 3u2 2 4u)
2t 2 1 8t 1 5
12. 25(2c2 2 d 2)
23u 2 1 5u 2 10
13. x2(2x 1 9)
210c 2 1 5d 2
14. 2q(3pq 1 4q4)
2x 3 1 9x 2
15. 8w(hk2 1 10h3m4 2 6k5w3)
6pq 2 1 8q 5
16. m2n3(24m2n2 2 2mnp 2 7)
8hk 2w 1 80h 3m 4w 2 48k 5w 4
17. 23s2y(22s4y2 1 3sy3 1 4)
24m 4n 5 2 2m 3n 4p 2 7m 2n 3
18. (c 1 2)(c 1 8)
c2
a2
19. (z 2 7)(z 1 4)
z 2 2 3z 2 28
21. (2x 2 3)(3x 2 5)
6x 2 2 19x 1 15
2 10a 1 25
22. (r 2 2s)(r 1 2s)
r2
6s 6y 3 2 9s3y 4 2 12s 2y
1 10c 1 16
20. (a 2 5)2
2
23. (3y 1 4)(2y 2 3)
4s 2
24. (3 2 2b)(3 1 2b)
6y 2 2 y 2 12
25. (3w 1 1)2
9 2 4b 2
©
Glencoe/McGraw-Hill
Lesson 5-2
4. (g 1 5) 1 (2g 1 7)
9w 2 1 6w 1 1
247
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-3
____________ PERIOD _____
Study Guide and Intervention
(continued)
Dividing Polynomials
Use Synthetic Division
a procedure to divide a polynomial by a binomial using coefficients of the dividend and
the value of r in the divisor x 2 r
Synthetic division
Use synthetic division to find (2x3 2 5x2 1 5x 2 2) 4 (x 2 1).
Step 1
Write the terms of the dividend so that the degrees of the terms are in
descending order. Then write just the coefficients.
2x 3 2 5x 2 1 5x 2 2
2
25
5 22
Step 2
Write the constant r of the divisor x 2 r to the left, In this case, r 5 1.
Bring down the first coefficient, 2, as shown.
1 2
25
5
22
25
2
23
5
22
25
2
23
5
23
2
22
25
2
23
5
23
2
22
2
0
2
Step 3
Step 4
Multiply the first coefficient by r, 1 ? 2 5 2. Write their product under the
second coefficient. Then add the product and the second coefficient:
25 1 2 5 2 3.
1 2
Multiply the sum, 23, by r: 23 ? 1 5 23. Write the product under the next
coefficient and add: 5 1 (23) 5 2.
1 2
2
2
Step 5
Multiply the sum, 2, by r: 2 ? 1 5 2. Write the product under the next
coefficient and add: 22 1 2 5 0. The remainder is 0.
1 2
2
Thus, (2x3 2 5x2 1 5x 2 2) 4 (x 2 1) 5 2x2 2 3x 1 2.
Exercises
Simplify.
1. (3x3 2 7x2 1 9x 2 14) 4 (x 2 2)
2. (5x3 1 7x2 2 x 2 3) 4 (x 1 1)
3x 2 2 x 1 7
5x 2 1 2x 2 3
3. (2x3 1 3x2 2 10x 2 3) 4 (x 1 3)
4. (x3 2 8x2 1 19x 2 9) 4 (x 2 4)
3
2x 2 2 3x 2 1
x 2 2 4x 1 3 1 }
x24
5. (2x3 1 10x2 1 9x 1 38) 4 (x 1 5)
6. (3x3 2 8x2 1 16x 2 1) 4 (x 2 1)
7
10
2x 2 1 9 2 }
x15
3x 2 2 5x 1 11 1 }
x21
7. (x3 2 9x2 1 17x 2 1) 4 (x 2 2)
8. (4x3 2 25x2 1 4x 1 20) 4 (x 2 6)
5
8
x 2 2 7x 1 3 1 }
x22
9. (6x3 1 28x2 2 7x 1 9) 4 (x 1 5)
4x 2 2 x 2 2 1 }
x26
10. (x4 2 4x3 1 x2 1 7x 2 2) 4 (x 2 2)
6
6x 2 2 2x 1 3 2 }
x15
x 3 2 2x 2 2 3x 1 1
265
11. (12x4 1 20x3 2 24x2 1 20x 1 35) 4 (3x 1 5) 4x 3 2 8x 1 20 1 }
3x 1 5
©
Glencoe/McGraw-Hill
252
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-3
____________ PERIOD _____
Skills Practice
Dividing Polynomials
Simplify.
12x 1 20
4
10c 1 6
2
2. }} 3x 1 5
3. }} 5y 2 1 2y 1 1
15y3 1 6y2 1 3y
3y
4. }} 3x 2 1 2 }
5. (15q6 1 5q2)(5q4)21
6. (4f 5 2 6f 4 1 12f 3 2 8f 2)(4f 2)21
1. } 5c 1 3
2
x
12x2 2 4x 2 8
4x
3f 2
2
1
q
3q 2 1 }2
f 3 2 } 1 3f 2 2
7. (6j 2k 2 9jk2) 4 3jk
8. (4a2h2 2 8a3h 1 3a4) 4 (2a2)
3a 2
2
2h 2 2 4ah 1 }
2j 2 3k
9. (n2 1 7n 1 10) 4 (n 1 5)
10. (d 2 1 4d 1 3) 4 (d 1 1)
n12
d13
11. (2s2 1 13s 1 15) 4 (s 1 5)
12. (6y2 1 y 2 2)(2y 2 1)21
3y 1 2
13. (4g2 2 9) 4 (2g 1 3)
Lesson 5-3
2s 1 3
14. (2x2 2 5x 2 4) 4 (x 2 3)
1
2g 2 3
2x 1 1 2 }
x23
u2 1 5u 2 12
u23
2x2 2 5x 2 4
x23
15. }}
16. }}
1
12
u181}
u23
2x 1 1 2 }
x23
17. (3v2 2 7v 2 10)(v 2 4)21
18. (3t4 1 4t3 2 32t2 2 5t 2 20)(t 1 4)21
10
3t 3 2 8t 2 2 5
3v 1 5 1 }
v24
y3 2 y2 2 6
y12
2x3 2 x2 2 19x 1 15
x23
19. }}
20. }}}
18
3
y 2 2 3y 1 6 2 }
y12
2x 2 1 5x 2 4 1 }
x23
21. (4p3 2 3p2 1 2p) 4 ( p 2 1)
22. (3c4 1 6c3 2 2c 1 4)(c 1 2)21
8
3
4p 2 1 p 1 3 1 }
p21
3c 3 2 2 1 }
c12
23. GEOMETRY The area of a rectangle is x3 1 8x2 1 13x 2 12 square units. The width of
the rectangle is x 1 4 units. What is the length of the rectangle? x 2 1 4x 2 3 units
©
Glencoe/McGraw-Hill
253
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-3
Practice
____________ PERIOD _____
(Average)
Dividing Polynomials
Simplify.
8
15r10 2 5r8 1 40r2
1. }}}
3r 6 2 r 4 1 }2
4
2. }}}
} 2 6k 2 1 }
2
3. (230x3y 1 12x2y2 2 18x2y) 4 (26x2y)
4. (26w3z4 2 3w2z5 1 4w 1 5z) 4 (2w2z)
5r
5.
r
6k2m 2 12k3m2 1 9m3 3k
2km
m
4
9m
2k
5
2w
5x 2 2y 1 3
2
3z
23wz 3 2 } 1 } 1 }2
(4a3
(28d 3k2
2
8a2
1
2
a2)(4a)21
6.
a
4
a 2 2 2a 1 }}
1
d
4
d 2k2
wz
2 4dk2)(4dk2)21
7d 2 1 }} 2 1
f 2 1 7f 1 10
f12
2x2 1 3x 2 14
x22
7. }} f 1 5
8. }} 2x 1 7
9. (a3 2 64) 4 (a 2 4) a 2 1 4a 1 16
2x3 1 6x 1 152
x14
10. (b3 1 27) 4 (b 1 3) b 2 2 3b 1 9
72
x13
3
11. }} 2x 2 2 8x 1 38
2x 1 4x 2 6
12. }} 2x 2 2 6x 1 22 2 }
13. (3w3 1 7w2 2 4w 1 3) 4 (w 1 3)
14. (6y4 1 15y3 2 28y 2 6) 4 (y 1 2)
x13
3
w13
26
y12
3w 2 2 2w 1 2 2 }}
6y 3 1 3y 2 2 6y 2 16 1 }}
15. (x4 2 3x3 2 11x2 1 3x 1 10) 4 (x 2 5)
16. (3m5 1 m 2 1) 4 (m 1 1)
5
m11
x3 1 2x 2 2 x 2 2
3m4 2 3m 3 1 3m 2 2 3m 1 4 2 }
17. (x4 2 3x3 1 5x 2 6)(x 1 2)21
18. (6y2 2 5y 2 15)(2y 1 3)21
24
x12
6
2y 1 3
x 3 2 5x 2 1 10x 2 15 1 }}
4x2 2 2x 1 6
2x 2 3
3y 2 7 1 }}
6x2 2 x 2 7
3x 1 1
19. }}
20. }}
12
2x 2 3
2x 1 2 1 }}
21. (2r3 1 5r2 2 2r 2 15) 4 (2r 2 3)
22. (6t3 1 5t2 2 2t 1 1) 4 (3t 1 1)
2
3t 1 1
r 2 1 4r 1 5
4p4 2 17p2 1 14p 2 3
2p 2 3
3
2p 1 3p 2 2 4p 1
6
3x 1 1
2x 2 1 2 }}
2t 2 1 t 2 1 1 }}
23. }}}
2h4 2 h3 1 h2 1 h 2 3
h 21
2
2h 2 h 1 3
24. }}}
2
1
25. GEOMETRY The area of a rectangle is 2x2 2 11x 1 15 square feet. The length of the
rectangle is 2x 2 5 feet. What is the width of the rectangle? x 2 3 ft
26. GEOMETRY The area of a triangle is 15x4 1 3x3 1 4x2 2 x 2 3 square meters. The
length of the base of the triangle is 6x2 2 2 meters. What is the height of the triangle?
5x 2 1 x 1 3 m
©
Glencoe/McGraw-Hill
254
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-4
____________ PERIOD _____
Study Guide and Intervention
Factoring Polynomials
Factor Polynomials
For any number of terms, check for:
greatest common factor
For two terms, check for:
Difference of two squares
a 2 2 b 2 5 (a 1 b)(a 2 b)
Sum of two cubes
a 3 1 b 3 5 (a 1 b)(a 2 2 ab 1 b 2)
Difference of two cubes
a 3 2 b 3 5 (a 2 b)(a 2 1 ab 1 b 2)
Techniques for Factoring Polynomials
For three terms, check for:
Perfect square trinomials
a 2 1 2ab 1 b 2 5 (a 1 b)2
a 2 2 2ab 1 b 2 5 (a 2 b)2
General trinomials
acx 2 1 (ad 1 bc)x 1 bd 5 (ax 1 b)(cx 1 d)
For four terms, check for:
Grouping
ax 1 bx 1 ay 1 by 5 x(a 1 b) 1 y(a 1 b)
5 (a 1 b)(x 1 y)
Example
Factor 24x2 2 42x 2 45.
First factor out the GCF to get 24x2 2 42x 2 45 5 3(8x2 2 14x 2 15). To find the coefficients
of the x terms, you must find two numbers whose product is 8 ? (215) 5 2120 and whose
sum is 214. The two coefficients must be 220 and 6. Rewrite the expression using 220x and
6x and factor by grouping.
8x2 2 14x 2 15 5 8x2 2 20x 1 6x 2 15
5 4x(2x 2 5) 1 3(2x 2 5)
5 (4x 1 3)(2x 2 5)
Group to find a GCF.
Factor the GCF of each binomial.
Distributive Property
Exercises
Factor completely. If the polynomial is not factorable, write prime.
1. 14x2y2 1 42xy3
14xy 2(x 1 3y)
4. x4 2 1
(x 2 1 1)(x 1 1)(x 2 1)
7. 100m8 2 9
(10m 4 2 3)(10m 4 1 3)
©
Glencoe/McGraw-Hill
2. 6mn 1 18m 2 n 2 3
(6m 2 1)(n 1 3)
5. 35x3y4 2 60x4y
5x 3y(7y 3 2 12x)
8. x2 1 x 1 1
3. 2x2 1 18x 1 16
2(x 1 8)(x 1 1)
6. 2r3 1 250
2(r 1 5)(r 2 2 5r 1 25)
9. c4 1 c3 2 c2 2 c
c(c 1 1)2 (c 2 1)
prime
257
Glencoe Algebra 2
Lesson 5-4
Thus, 24x2 2 42x 2 45 5 3(4x 1 3)(2x 2 5).
NAME ______________________________________________ DATE
5-4
____________ PERIOD _____
Study Guide and Intervention
(continued)
Factoring Polynomials
Simplify Quotients
In the last lesson you learned how to simplify the quotient of two
polynomials by using long division or synthetic division. Some quotients can be simplified by
using factoring.
Example
8x2 1 11x 1 12
2x 2 13x 2 24
Simplify }}
.
2
8x2 1 11x 1 12
(2x 1 3)( x 1 4)
}}
5 }}
2x2 2 13x 2 24
(x 2 8)(2x 1 3)
x14
5}
x28
Factor the numerator and denominator.
3
2
Divide. Assume x Þ 8, 2 } .
Exercises
Simplify. Assume that no denominator is equal to 0.
x2 2 7x 1 12
x 2x26
1. }}
2
x24
}
x12
x2 1 x 2 6
x 2 7x 1 10
4. }}
2
x13
}
x25
4x2 1 4x 2 3
2x 2 x 2 6
7. }}
2
2x 2 1
}
x22
4x2 1 16x 1 15
2x 1 x 2 3
10. }}
2
2x 1 5
}
x21
x2 2 81
2x 2 23x 1 45
13. }}
2
x19
}
2x 2 5
4x2 2 4x 2 3
8x 1 1
16. }}
3
2x 2 3
}}
2
4x 2 2x 1 1
©
Glencoe/McGraw-Hill
x2 1 6x 1 5
2x 2 x 2 3
2. }}
2
x15
}
2x2 3
x2 2 11x 1 30
x 2 5x 2 6
3. }}
2
x25
}
x11
2x2 1 5x 2 3
4x 1 11x 2 3
5. }}
2
2x 2 1
}
4x 2 1
5x2 1 9x 2 2
x 1 5x 1 6
6. }}
2
5x 2 1
}
x13
6x2 1 25x 1 4
x 1 6x 1 8
8. }}
2
6x 1 1
}
x12
x2 2 7x 1 10
3x 2 8x 2 35
9. }}
2
x22
}
3x 1 7
3x2 1 4x 2 15
2x 1 3x 2 9
11. }}
2
3x 2 5
}
2x 2 3
x2 2 14x 1 49
x 2 2x 2 35
12. }}
2
x27
}
x15
7x2 1 11x 2 6
x 24
14. }}
2
7x 2 3
}
x22
4x2 2 12x 1 9
2x 1 13x 2 24
15. }}
2
2x 2 3
}
x18
y3 2 64
3y 2 17y 1 20
17. }}
2
y 2 1 4y 1 16
}}
3y 2 5
258
27x3 2 8
9x 2 4
18. }}
2
9x 2 1 6x 1 4
}}
3x 1 2
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-4
____________ PERIOD _____
Skills Practice
Factoring Polynomials
Factor completely. If the polynomial is not factorable, write prime.
1. 7x2 2 14x
2. 19x3 2 38x2
19x 2(x 2 2)
7x(x 2 2)
3. 21x3 2 18x2y 1 24xy2
4. 8j 3k 2 4jk3 2 7
3x(7x2 2 6xy 1 8y 2)
5. a2 1 7a 2 18
prime
6. 2ak 2 6a 1 k 2 3
(a 1 9)(a 2 2)
7. b2 1 8b 1 7
(2a 1 1)(k 2 3)
8. z2 2 8z 2 10
(b 1 7)(b 1 1)
9. m2 1 7m 2 18
prime
10. 2x2 2 3x 2 5
(m 2 2)(m 1 9)
11. 4z2 1 4z 2 15
(2x 2 5)(x 1 1)
12. 4p2 1 4p 2 24
(2z 1 5)(2z 2 3)
13. 3y2 1 21y 1 36
4(p 2 2)(p 1 3)
14. c2 2 100
3(y 1 4)(y 1 3)
15. 4f 2 2 64
(c 1 10)(c 2 10)
16. d 2 2 12d 1 36
(d 2 6)2
4(f 1 4)(f 2 4)
18. y2 1 18y 1 81
Lesson 5-4
17. 9x2 1 25
(y 1 9)2
prime
19. n3 2 125
20. m4 2 1
(n 2 5)(n 2 1 5n 1 25)
(m 2 1 1)(m 2 1)(m 1 1)
Simplify. Assume that no denominator is equal to 0.
x2 1 7x 2 18 x 2 2
}
21. }}
x2 1 4x 2 45 x 2 5
x25
2
x 2 10x 1 25 }
23. }}
2
x
x 2 5x
©
Glencoe/McGraw-Hill
x2 1 4x 1 3 x 1 1
}
22. }}
x2 1 6x 1 9 x 1 3
x2 1 6x 2 7 x 2 1
}
24. }}
x27
x2 2 49
259
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-4
Practice
____________ PERIOD _____
(Average)
Factoring Polynomials
Factor completely. If the polynomial is not factorable, write prime.
1. 15a2b 2 10ab2
5ab(3a 2 2b)
4. 2x3y 2 x2y 1 5xy2 1 xy3
xy(2x 2 2 x 1 5y 1 y 2)
7. y2 1 20y 1 96
(y 1 8)(y 1 12)
10. 6x2 1 7x 2 3
(3x 2 1)(2x 1 3)
13. r3 1 3r2 2 54r
r(r 1 9)(r 2 6)
16. x3 1 8
2. 3st2 2 9s3t 1 6s2t2
3st(t 2 3s 2 1 2st)
19. 8m3 2 25 prime
xy(3x 2y 2 2x 1 5)
5. 21 2 7t 1 3r 2 rt
6. x2 2 xy 1 2x 2 2y
(7 1 r)(3 2 t)
(x 1 2)(x 2 y)
9. 6n2 2 11n 2 2
8. 4ab 1 2a 1 6b 1 3
(2a 1 3)(2b 1 1)
11. x2 2 8x 2 8
(6n 1 1)(n 2 2)
12. 6p2 2 17p 2 45
prime
(2p 2 9)(3p 1 5)
14. 8a2 1 2a 2 6
15. c2 2 49
2(4a 2 3)(a 1 1)
17. 16r2 2 169
(x 1 2)(x 2 2 2x 1 4)
3. 3x3y2 2 2x2y 1 5xy
(c 2 7)(c 1 7)
18. b4 2 81
(4r 1 13)(4r 2 13)
(b 2 1 9)(b 1 3)(b 2 3)
20. 2t3 1 32t2 1 128t 2t(t 1 8)2
21. 5y5 1 135y2 5y 2(y 1 3)(y 2 2 3y 1 9) 22. 81x4 2 16 (9x 2 1 4)(3x 1 2)(3x 2 2)
Simplify. Assume that no denominator is equal to 0.
x2 2 16
x14
}
23. }}
x2 1 x 2 20 x 1 5
x2 2 16x 1 64 x 2 8
}
24. }}
x2 1 x 2 72 x 1 9
3(x 1 3)
x 2 27 x 1 3x 1 9
2
3x 2 27 }}
25. }}
2
3
26. DESIGN Bobbi Jo is using a software package to create a
drawing of a cross section of a brace as shown at the right.
Write a simplified, factored expression that represents the
area of the cross section of the brace. x(20.2 2 x) cm2
cm
12 cm
x
x
cm
8.2 cm
27. COMBUSTION ENGINES In an internal combustion engine, the up
and down motion of the pistons is converted into the rotary motion of
the crankshaft, which drives the flywheel. Let r1 represent the radius
of the flywheel at the right and let r2 represent the radius of the
crankshaft passing through it. If the formula for the area of a circle
is A 5 pr2, write a simplified, factored expression for the area of the
cross section of the flywheel outside the crankshaft. p (r1 2 r2)(r1 1 r2)
©
Glencoe/McGraw-Hill
260
r1
r2
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-5
____________ PERIOD _____
Skills Practice
Roots of Real Numbers
Use a calculator to approximate each value to three decimal places.
w 15.166
1. Ï230
2. Ï38
w 6.164
3. 2Ï152
w 212.329
4. Ï5.6
w 2.366
3
5. Ï88
w 4.448
4
7. 2Ï0.34
w 20.764
3
6. Ï2222
w 26.055
5
8. Ï500
w 3.466
Simplify.
9. 6Ï81
w 69
10. Ï144
w 12
11. Ïw
(25)2 5
12. Ïw
252 not a real number
13. Ï0.36
w 0.6
14. 2
Îã 2}23
4
}
9
3
16. 2Ï27
w 23
3
18. Ï32
w 2
19. Ï81
w 3
4
20. Ïw
y2 | y |
21. Ïw
125s3 5s
22. Ïw
64x6 8| x 3|
23. Ï227a
w6w 23a 2
24. Ïw
m8n4 m 4n 2
25. 2Ïw
100p4w
q2 210p 2| q |
26. Ïw
16w4v8w 2| w | v 2
27. Ïw
(23c)4 9c 2
28. Ïw
(a 1 bw
)2 | a 1 b |
17. Ï0.064
w 0.4
3
3
©
Glencoe/McGraw-Hill
3
5
Lesson 5-5
15. Ï28
w 22
4
265
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-6
____________ PERIOD _____
Skills Practice
Radical Expressions
w 2Ï6
w
1. Ï24
Lesson 5-6
Simplify.
2. Ï75
w 5Ï3
w
3
4
3. Ï16
w 2Ï2
w
4. 2Ï48
w 22 Ï3
w
5. 4Ïw
50x5 20x 2Ï2x
w
6. Ïw
64a4b4w 2| ab | Ï4
w
3
7.
3
1
8
2 } d 2f 5
3
Îã
3
}
7
4
4
d f
Îã 2}12 f Ïw
9. 2
11.
4
2 2
Ï21
w
2}
8.
10.
7
g Ï10gz
w
Îã }}
5z
2g3
}}
5z
Îã }56 |s |Ïtw
25
} s2t
36
Îã
3
3
w
2 Ï6
} }
9
3
12. (3Ï3
w )(5Ï3
w ) 45
13. (4Ï12
w )(3Ï20
w ) 48Ï15
w
14. Ï2
w 1 Ï8
w 1 Ï50
w 8Ï2
w
15. Ï12
w 2 2Ï3
w 1 Ï108
w 6Ï3
w
16. 8Ï5
w 2 Ï45
w 2 Ï80
w
18. (2 1 Ï3
w )(6 2 Ï2
w ) 12 2 2Ï2
w 1 6Ï3
w2
Ï6
w
19. (1 2 Ï5
w )(1 1 Ï5
w ) 24
20. (3 2 Ï7
w )(5 1 Ï2
w ) 15 1 3Ï2
w 2 5Ï7
w2
Ï14
w
21. (Ï2
w 2 Ï6
w ) 8 2 4Ï3
w
22. } }}
12 2 4Ï2
w
4
7
3 1 Ï2
w
24. } }}
17. 2Ï48
w 2 Ï75
w 2 Ï12
w
2
23. } }}
©
Glencoe/McGraw-Hill
Ï3
w
Ï5
w
21 1 3Ï2
w
3
47
7 2 Ï2
w
40 1 5Ï6
w
5
58
8 2 Ï6
w
271
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-7
____________ PERIOD _____
Skills Practice
Rational Exponents
Write each expression in radical form.
1
}}
1
}}
6
Ï3
w
1. 3 6
2
Ïw
122 or (Ï12
w)
2
}}
3
3. 12 3
5
Ï8
w
2. 8 5
3
3
}}
5
4. (s3) 5 sÏw
s4
5. Ï51
w 51
1
}}
2
3
}}
7. Ïw
153 15 4
4
Lesson 5-7
Write each radical using rational exponents.
1
}}
3
3
6. Ï37
w 37
1
}}
1
}}
2
}}
8. Ïw
6xy2 6 3 x 3 y 3
3
Evaluate each expression.
1
}}
1
}}
9. 32 5 2
1
2}3}
11. 27
10. 81 4 3
1
}
3
3
}}
4
}}
13. 16 2 64
1
}}
1
2
1
12. 42}2} }
14. (2243) 5 81
5
}}
15. 27 3 ? 27 3 729
8
}
27
3
}}
2
1 49 2
16. }
Simplify each expression.
12
}}
3
}}
17. c 5 ? c 5 c 3
1 2
1
}}
2
3
19. q
6
2 }11}
21. x
1
2} }
q
3
}}
2
5
}}
11
x
}
x
12
©
Ï2
w
Glencoe/McGraw-Hill
16
}}
4
}}
p5
}
p
1
2}5}
20. p
2
}}
x3
22. }
1
}}
x4
1
}}
y 2 y4
23. }
}
1
}}
y
y4
25. Ï64
w
2
}}
18. m 9 ? m 9 m 2
x
1
}}
5
}}
12
2
}}
n3
n3
24. }
}
1
1
}}
}}
n6 ? n2 n
4
26. Ïw
49a8b2w | a | Ï7b
w
8
277
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-7
Practice
____________ PERIOD _____
(Average)
Rational Exponents
Write each expression in radical form.
1
}}
2
}}
4
}}
2. 6 5
1. 5 3
2
Ïw
62 or (Ï6
w)
3
5
Ï5
w
2
}}
4. (n3) 5
3. m 7
4
Ïw
m4 or (Ïm
w)
5
7
7
5
n Ïn
w
Write each radical using rational exponents.
7. Ïw
27m6n4w
4
5. Ï79
w
1
}}
4
}}
1
}}
79 2
8. 5Ïw
2a10b
3
6. Ï153
w
1
}}
3m 2n 3
153 4
1
}}
5 ? 2 2 |a 5 | b 2
Evaluate each expression.
1
}}
10. 1024
1
64
3
12. 2256
1
125
216
15. }
1
}
4
1
2}5}
9. 81 4 3
2}4}
13. (264)
2
1
}}
343
4
}}
1 21
}}
64 3 16
16. }
}
2
1
}}
1
17. 25 2 264
49
}}
3
1
}
32
14. 27 3 ? 27 3 243
2
25
}
36
2
}}
3
1
}
16
2
2}3}
2}
5
2}3}
11. 8
2}3}
Simplify each expression.
4
}}
7
3
}}
7
18. g ? g
3
2}5}
22. b
g
2
}}
5
b
}
b
10
26. Ïw
85 2Ï2
w
3
}}
4
13
}}
4
19. s ? s
3
}}
5
q
23. }2 q
1
s4
1
20. u
1
}}
5
2
4
2}3} 2}5}
u
4
}}
15
11
}}
12
2
}}
3
t
t
24. }
}
1
3
}}
5
}}
q
2}4}
5t 2 ? t
27. Ï12
w ? Ïw
123
28. Ï6
w ? 3Ï6
w
12Ï12
w
3Ï6
w
5
4
10
4
5
2
5
4
2}
1
}}
y2
}
y
1
2}2}
21. y
1
}}
1
}}
2
2z 1 2z 2
2z
}}
25. }
1
z21
}}
z2 2 1
a
aÏ3b
w
29. } }
Ï3b
w
3b
30. ELECTRICITY The amount of current in amperes I that an appliance uses can be
1
}}
1 2
P
R
calculated using the formula I 5 } 2 , where P is the power in watts and R is the
resistance in ohms. How much current does an appliance use if P 5 500 watts and
R 5 10 ohms? Round your answer to the nearest tenth. 7.1 amps
1
}}
31. BUSINESS A company that produces DVDs uses the formula C 5 88n 3 1 330 to
calculate the cost C in dollars of producing n DVDs per day. What is the company’s cost
to produce 150 DVDs per day? Round your answer to the nearest dollar. $798
©
Glencoe/McGraw-Hill
278
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-8
____________ PERIOD _____
Study Guide and Intervention
Radical Equations and Inequalities
Solve Radical Equations The following steps are used in solving equations that have
variables in the radicand. Some algebraic procedures may be needed before you use these
steps.
1
2
3
4
Isolate the radical on one side of the equation.
To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.
Solve the resulting equation.
Check your solution in the original equation to make sure that you have not obtained any extraneous roots.
Example 1
Example 2
Solve 2Ïw
4x 1 8
w 2 4 5 8.
2Ïw
4x 1 8 2 4 5 8
2Ïw
4x 1 8 5 12
Ïw
4x 1 8 5 6
4x 1 8 5 36
4x 5 28
x57
Check
Solve Ïw
3x 1 1
w 5 Ï5x
w 2 1.
Ïw
3x 1 1 5 Ï5x
w21
Original equation
3x 1 1 5 5x 2 2Ïw
5x 1 1 Square each side.
2Ï5x
w 5 2x
Simplify.
Ï5x
w5x
Isolate the radical.
5x 5 x2
Square each side.
2
x 2 5x 5 0
Subtract 5x from each side.
x(x 2 5) 5 0
Factor.
x 5 0 or x 5 5
Check
Ïw
3(0) 1w
1 5 1, but Ï5(0)
w 2 1 5 21, so 0 is
not a solution.
Ïw
3(5) 1w
1 5 4, and Ï5(5)
w 2 1 5 4, so the
solution is x 5 5.
Original equation
Add 4 to each side.
Isolate the radical.
Square each side.
Subtract 8 from each side.
Divide each side by 4.
2Ïw
4(7) 1w
82408
2Ï36
w2408
2(6) 2 4 0 8
858
The solution x 5 7 checks.
Exercises
Solve each equation.
1. 3 1 2xÏ3
w55
Ï3
w
}
3
4. Ïw
52x2456
295
7. Ï21
w 2 Ïw
5x 2 4 5 0
5
15
3
8
Glencoe/McGraw-Hill
3. 8 1 Ïw
x1152
no solution
5. 12 1 Ïw
2x 2 1 5 4
no solution
6. Ïw
12 2 x
w50
12
8. 10 2 Ï2x
w55
12.5
10. 4Ïw
2x 1 11
w 2 2 5 10
©
2. 2Ïw
3x 1 4 1 1 5 15
9. Ïw
x2 1 7x
w 5 Ïw
7x 2 9
no solution
11. 2Ïw
x 1 11 5 Ïw
x 1 2 1 Ïw
3x 2 6
14
12. Ïw
9x 2 11
w5x11
3, 4
281
Glencoe Algebra 2
Lesson 5-8
Step
Step
Step
Step
NAME ______________________________________________ DATE
5-8
____________ PERIOD _____
Skills Practice
Radical Equations and Inequalities
Solve each equation or inequality.
1
25
3. 5Ïjw 5 1 }
1
}}
2. Ïx
w 1 3 5 7 16
1
}}
4. v 2 1 1 5 0 no solution
3
5. 18 2 3y 2 5 25 no solution
6. Ï2w
w 5 4 32
7. Ïw
b 2 5 5 4 21
8. Ïw
3n 1 1
w55 8
3
9. Ïw
3r 2 6 5 3 11
11. Ïw
k 2 4 2 1 5 5 40
1
}}
Lesson 5-8
w 5 5 25
1. Ïx
10. 2 1 Ïw
3p 1 7
w56 3
5
2
1
}}
12. (2d 1 3) 3 5 2 }
1
}}
13. (t 2 3) 3 5 2 11
14. 4 2 (1 2 7u) 3 5 0 29
15. Ïw
3z 2 2 5 Ïw
z 2 4 no solution
16. Ïw
g 1 1 5 Ïw
2g 2 7
w 8
17. Ïw
x 2 1 5 4Ïw
x 1 1 no solution
18. 5 1 Ïw
s23#6 3#s#4
19. 22 1 Ïw
3x 1 3 , 7 21 , x , 26
20. 2Ïw
2a 1 4
w $ 26 22 # a # 16
21. 2Ïw
4r 2 3 . 10 r . 7
22. 4 2 Ïw
3x 1 1 . 3 2 } , x , 0
23. Ïw
y 1 4 2 3 $ 3 y $ 32
24. 23Ïw
11r 1w
3 $ 215 2 } # r # 2
©
Glencoe/McGraw-Hill
1
3
3
11
283
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-9
____________ PERIOD _____
Skills Practice
Complex Numbers
Simplify.
w 6i
1. Ï236
2. Ï2196
w 14i
3. Ïw
281x6 9 | x 3 | i
4. Ï223
w ? Ï246
w 223Ï2
w
5. (3i)(22i)(5i) 30i
6. i 11 2i
7. i 65 i
8. (7 2 8i) 1 (212 2 4i) 25 2 12i
10. (10 2 4i) 2 (7 1 3i) 3 2 7i
11. (2 1 i)(2 1 3i) 1 1 8i
12. (2 1 i)(3 2 5i) 11 2 7i
13. (7 2 6i)(2 2 3i) 24 2 33i
14. (3 1 4i)(3 2 4i) 25
26 2 8i
8 2 6i
15. } }
3
3i
3 1 6i
3i
16. } }
10
Lesson 5-9
9. (23 1 5i) 1 (18 2 7i) 15 2 2i
4 1 2i
Solve each equation.
17. 3x2 1 3 5 0 6i
18. 5x2 1 125 5 0 65i
19. 4x2 1 20 5 0 6i Ï5
w
20. 2x2 2 16 5 0 64i
21. x2 1 18 5 0 63i Ï2
w
22. 8x2 1 96 5 0 62i Ï3
w
Find the values of m and n that make each equation true.
23. 20 2 12i 5 5m 1 4ni 4, 23
24. m 2 16i 5 3 2 2ni 3, 8
25. (4 1 m) 1 2ni 5 9 1 14i 5, 7
26. (3 2 n) 1 (7m 2 14)i 5 1 1 7i 3, 2
©
Glencoe/McGraw-Hill
289
Glencoe Algebra 2
NAME ______________________________________________ DATE
5-9
Practice
____________ PERIOD _____
(Average)
Complex Numbers
Simplify.
w 7i
1. Ï249
2. 6Ï212
w 12i Ï3
w
3. Ï2121
ww
s8 11s 4i
4. Ï236a
w3w
b4
5. Ï28
w ? Ï232
w
6. Ï215
w ? Ï225
w
6| a|
b2i Ïa
w
25Ï15
w
216
8. (7i)2(6i)
7. (23i)(4i)(25i)
260i
9. i 42
2294i
10. i 55
21
11. i 89
12. (5 2 2i) 1 (213 2 8i)
28 2 10i
i
2i
13. (7 2 6i) 1 (9 1 11i)
14. (212 1 48i) 1 (15 1 21i)
16 1 5i
16. (28 2 4i) 2 (10 2 30i)
3 1 69i
14 1 16i
7 2 8i
18. (8 2 11i)(8 2 11i)
52
257 2 176i
20. (7 1 2i)(9 2 6i)
23 2 14i
2
22. } }}
113
238 1 45i
17. (6 2 4i)(6 1 4i)
18 1 26i
19. (4 1 3i)(2 2 5i)
15. (10 1 15i) 2 (48 2 30i)
25 1 6i
6 1 5i
21. } }
2
22i
75 2 24i
71i
32i
23. } }
5
2 2 4i
1 1 3i
24. } 21 2 i
22i
Solve each equation.
25. 5n2 1 35 5 0 6i Ï7
w
26. 2m2 1 10 5 0 6i Ï5
w
27. 4m2 1 76 5 0 6i Ï19
w
28. 22m2 2 6 5 0 6i Ï3
w
29. 25m2 2 65 5 0 6i Ï13
w
30. } x2 1 12 5 0 64i
3
4
Find the values of m and n that make each equation true.
31. 15 2 28i 5 3m 1 4ni 5, 27
32. (6 2 m) 1 3ni 5 212 1 27i 18, 9
33. (3m 1 4) 1 (3 2 n)i 5 16 2 3i 4, 6
34. (7 1 n) 1 (4m 2 10)i 5 3 2 6i 1, 24
35. ELECTRICITY The impedance in one part of a series circuit is 1 1 3j ohms and the
impedance in another part of the circuit is 7 2 5j ohms. Add these complex numbers to
find the total impedance in the circuit. 8 2 2j ohms
36. ELECTRICITY Using the formula E 5 IZ, find the voltage E in a circuit when the
current I is 3 2 j amps and the impedance Z is 3 1 2j ohms. 11 1 3j volts
©
Glencoe/McGraw-Hill
290
Glencoe Algebra 2