Download ALGEBRA I- MATH 006 HOMEWORK I SOLUTIONS 1

ALGEBRA I- MATH 006
HOMEWORK I
SOLUTIONS
1. Write each statement as an inequality.
a) x is positive
b) x is less than or equal to 1.
Solution: a) x > 0 b) x ≤ 1
2. Graph the numbers x on the real number line.
a) x ≥ −2
b) x < 4
Solution:
-3
-2
1
4
3. If P = −3 and Q = 2, compute the distance between the points P and
Q.
Solution: d(P, Q) = |2 − (−3)| = |2 + 3| = 5
4. Find the value of each expression if x = −2 and y = 3.
a) |x + y| b) ||4x| − |5y|| c) |y|
d) 2x−3
y
y
Solution: a) 1 b) 7 c) 1 d) −7
3
5. Determine the domain of the variable x in each expression.
4
x−2
a) x−5
b) x+4
Solution: a) {x : x 6= 5} b) {x : x 6= −4}
6. Simplify each expression.
q
√
a) (−3)2 b) −3−2 c) 3−6 · 34 d) 81 e) (−3)2
Solution: a) 9 b) − 91 c) 19 d) 9 e) |−3| = 3
1
7. Simplify each expression. Express the answer so that all exponents are
positive.
−2
−1
a) (8x3 )2 b) (x−1 y)3 c) xxy2y d) ( 3x
)−2
4y −1
Solution: a) 64x6 b)
y3
x3
c)
1
x3 y
d)
16x2
9y 2
8. The lengths of the legs of a right triangle are given. Find the hypotenuse.
a = 6, b = 8
√
Solution: c2 = a2 + b2 = 36 + 64 = 100, c = 100 = 10
9. The lengths of the sides of a triangle are given. Determine which is a
right triangle, and identify the length of the hypotenuse.
a) 3, 4, 5 b) 2, 2, 3
Solution: a) Since 52 = 32 + 42 , the length of hyp. is 5.
b) No
10. Find the area A of a rectangle with length 4 inches and width 2 inches.
Solution: A =(length)·(width)= 4 · 2 = 8 square inches
11. Find the area A and circumference C of a circle of radius 5 meters.
Solution: A = πr2 = π · 52 = 25π C = 2πr = 2π · 5 = 10π
12. Add, subtract, or multiply as indicated. Express your answer as a
single polynomial in standard form.
a) (x2 + 4x + 5) + (3x − 3) b) (x + 1)(x2 + 2x − 4)
Solution: a) x2 + 7x + 2 b) x3 + 3x2 − 2x − 4
13. Multiply the following polynomials. Express your answer as a single
polynomial in standard form.
a)(x + 2)(x + 4) b) (x − 7)(x + 7) c) (x + 4)2 d) (x − 4)2
e) (3x + y)(3x − y)
Solution: a) x2 + 6x + 8 b) x2 − 49 c) x2 + 8x + 16
d) x2 − 8x + 16 e) 9x2 − y 2
2
14. Find the quotient and remainder.
a) 4x3 − 3x2 + x + 1 divided by x + 2
b) 3x3 − x2 + x + 1 divided by x2
Solution: a) quotient= 4x2 − 11x + 23, remainder= −45
b) quotient= 3x − 1, remainder= x + 1
15. Factor each polynomial.
a) 3x + 6 b) 2x2 − 2x
Solution: a) 3(x + 2) b) 2x(x − 1)
16. Factor each polynomial.
a) x2 − 1 b) 25x2 − 4 c) x2 + 4x + 4 d) x2 − 10x + 25
Solution: a) (x − 1)(x + 1) b) (5x − 2)(5x + 2) c) (x + 2)2
d) (x − 5)2
17. Factor each polynomial.
a) 3x2 + 4x + 1 b) 3x2 − 2x − 8
Solution: a) (3x + 1)(x + 1) b) (3x + 4)(x − 2)
18. Factor each polynomial.
a) x3 − 27 b) x3 + 27
Solution: a) (x − 3)(x2 + 3x + 9)
b) (x + 3)(x2 − 3x + 9)
19. Reduce each rational expression to lowest terms.
2
b) xx2 +4x−5
a) 3x+9
x2 −9
−2x+1
3
Solution: a) x−3
b) x+5
x−1
20. Perform the indicated operation and simplify. Leave your answer in
factored form.
6x
x
x2 −4
·
b)
a) 3x+6
3x−9
2
2
5x
x −4
2x+4
Solution: a)
3
5x(x−2)
b)
4x
(x−2)(x−3)
3
21. Perform the indicated operation and simplify. Leave your answer in
factored form.
4
2
4
x
− x+2
c) x−2
+ 2−x
a) x2 + 25 b) x−1
2(x+5)
Solution: a) x+5
b) (x−1)(x+2)
c) 4−x
2
x−2
22. Perform the indicated operation and simplify. Leave your answer in
factored
form.
1+ 1
a) 1− x1 b) 1 − 1−1 1
x
Solution: a)
x+1
x−1
x
b)
−1
x−1
23. Simplify each expression. Assume that all variables are positive when
they
√ √
√ appear.√
a) 3 27 b) 3 −8x4 c) q5x 20x3
q
√
√
3
Solution: a) 33 = 3 b) 3 (−2)3 x3 x = −2x 3 x c) 102 (x2 )2 = 10x2
1
3
3
24. Simplify each expression. a) 4 2 b) (−27)q
√ 3
√
3
3
Solution: a) ( 4) = 2 = 8 b) −27 = 3 (−3)3 = −3
25. Simplify each expression. Express your answer so that only positive
exponents occur.
2
1
a) x 3 x 2 x
−1
4
1
b)
11
Solution: a) x 12
2
(x2 y) 3 (xy 2 ) 3
2 2
x3 y3
2
3
b) x y
26. Rationalize √
the denominator of each expression.
1
√
a) 2 b) 5−√3 2
Solution: a)
√1
2
·
√
√2
2
√
=
2
2
b)
√
3
√
5− 2
4
·
√
5+√2
5+ 2
=
√ √
(5+ 2) 3
23