Selected J \
values are not allowed. 22. A rational expression involves a
ratio. 23. One is decreasing as the other increases.Lesson 11-1Go t I t? 1a. §, a * 0 b.pp. 664-669
9cL2-- d #
2d + 4' -2 c. 1 n * f
d. 13c, none 2a. x # - 2 , x * 4 b. — 3 —; a * 12z+ 3; z * - 2 , z * - I d. ^ 4 ; c * - 3 , c * - 2
-1; x # 2.5 b. -y -4 ;y * 4 c. + *
d * I d. --3 ” 2 z + 2 ' 2 ^ “ 1 4 a - 1 2 x + 4 ** N o ' ^ m u s t
be greater than 2-7T in order for the value of a to be
greater than 0. If to is less than or equal to 277 , then a will
be negative, and length cannot be negative.
Lesson Check 1. 3; x # -3 2. f, x # -3, x * 5- 4x 4a. No, the expression is not the ratio of two
polynomials, b. Yes, the expression is the ratio of two
polynomials. 5. If the denominator contains a polynomial,
there may be values of the variable that make the
denominator equal to zero, and division by zero is undefined. - The only way the rational expression is not in simplest
form is if the numerator and the denominator are equal.
7a. yes, 3 - x = - ( x - 3) b. no, 2 - y = - ( y - 2)
Ex e r c i se s 9. x 0 11. \,p 12 13. x 0
15.^4./^ ±4 17. ±7 19.^f, |
m -4, m -2 21. to + 3, to -5 23. -1, n * | - -2, m 2 27. ^ 5 - ^ ±5 29- w+ 1
- r i r - r + “5 33‘ f - 2. f J
- z3 z 0 37. - r i r i a - 3 , a -
c(3c + 5)
39- " W. ^ - f , c 2 41. No, y= x + 3
defined for x = - 3 but x - 3 is. 43. The student
canceled terms instead of factors;
*2 2 X2x = r i x 2) = 4 5 - Answers may vary.
2
X2 - 9is no tSample: (x_ 4)(x + 3) -r*. 5w + 6 — 47. 5 w 49. 3 + 3b 45.
a * -4to, a * 2to 51. Sometimes; it is true for all
values of to except 0. 53. Sometimes; it is true for all
values of a except - 1.Lesson 11-2Go t It? 1a. ri y * 0 b.(x - 2)(x - 3)x(x 4- 1)pp. 670-676, x * 3, x * 2
2a. 3x(x + 1) b. Yes, but you will have to simplify the
resulting expression. 3a. (x - 7)(3x - 2)
b. (x + 1)(x + 3) 4a. z2 + 2z- 1Lesso n Ch eck 1. ^ 2. (2* + 5 j(x 5)3. 3k2(k + 1)
4x
(x + 7)(2x + 3)r (a- 2)2 r5. — 3 ^— p , 6. x 2 7. no;b _ a a 1 _ a a_ to _ o. £ —
„ - h ' c ~ b c~ be where h a ' c a b- The procedures are the same, but when you multiply
rational expressions, there may be values of the variables
for which the rational expressions are not defined. - The variables to, c, and d appear in the denominators,
and division by 0 is not defined. 10a. Write the product of
the rational expression and the polynomial, factor, divide
out common factors, and write the product in factored
form. b. Rewrite the quotient of the rational expression
and the polynomial as the product of the rational
expression and the reciprocal of the polynomial. Factor the
numerators and denominators, divide out common factors,
and write the answer in factored form.
Ex e r c i se s 1 1. ^ 13 2c(c +^2 )
49.(r + 2)(r - 2)40 2 x (x - 1)
3a5 1 3 ' 3(x + 1) 17.(x-2 r
1)(x- 2)21.
27.f — 4 23. 4(f+ 1)(f + 2)
(b - 1 ) ( b + 4 ) x + 1c2
b - 1
3(b + 1)
1- 6 35.
43. 18 45
3
1 37.
129.
32
4n + 5 39.7k-^11 15
2(3g+ 1)
2(x + 1)47.g(3g-- f+ 3 53, 3f — 555.
1)
2
Mz+^ 0) It 2 x — 3- $88.71 59. $518,011.65 61. The student forgot to
rewrite the divisor as its reciprocal before canceling.
3 a. ( a + 2 ) 2 3 a a - 4 3 a ( a - 4 )
a + 2 a - (^4) (a + 2 )3
- 0, 4, and -
r03. r 2 m2(m + 2)
" a + 2 ( a + 2 ) 2
4 make the denominators equal 0.(m -1 )(m + 4) 67. '^1 69. (2 a + 3 b )(a + 2 b )(5a + b)(2a - 3b)71. G- 0 = —16f2 + 35t + 2.5 or 16f2 — 35f — 2.5 = 0;
use the quadratic formula to solve for t.
t =35 ± V 3 5 2 + 4 • 16 • 32 2.5 _ 35 ± 37.2 32 f « 2.26 s - I m 275. jp l—. a 0. a
76.2c- 9 2c + 8 'c - 4 , c* 4.5 77. 2x2 + 10x + 12 - -3n2 + 11n + 20 79. 6a3 - 21a2 + 2a - 7
Lesson 11-3 pp. 678-683Go t It? 1a. 2a + 5 + ^ b. to - | +^3
c. 2c3 + 3c + 1 2. 2m - 3 3a. q3 + q2 + 2q + 3
b. to2 - 3h + 5 - r i l 4a‘ 2 7 - T + 3(3y + 4)^55
b. 3a + 1 - r i r c - Check whether
(2x - 3)(2x + 2) - 7 equals 4x2 - 10x - 1.
Lesso n Ch eck 1. 4m + 2 - ^ ^- 20c + 43 + 3. 5n2 - 4n +1" 4. 3a - 5
5. Both processes involve dividing, multiplying, and
subtracting, then "bringing down," and repeating as
needed. When dividing polynomials you may need to