Download math 209 final

1.
Write each expression with positive exponents only and then simplify:
a. 2 –1 + 4 –1
1+1=2+3=5
2 4 4 4 5
b. ( 2/5 ) -3
(5/2)3 = 53÷23 = 125/8
2. Add or subtract as indicated:
a. ( - 6X3 + 7 X2 – 9X + 3) + (14X3 + 3X2 –11X –7)
-6X3 + 14X3 + 7X2 + 3X2 – 9X – 11X + 3 – 7
18X3 + 10X2 – 20X – 4
b. (13X4 - 8X3 + 2X2 ) – ( 5X4 –3X3 + 2X2 –6)
13X4 – 5X4 – 8X3 + 3X3 + 2x2 – 2X2 + 6
8X4 – 5X3 + 6
3. Find each product:
a. 3y2 (-7y2 + 3y –6) –21y4 + 9y3 – 18y2
b. (3y –2 ) ( 4y2 + 3y –5 ) 12y3 – 8y2 + 9y2 – 6y – 15y + 10
12y3 + y2 – 21y + 10
4. Divide:
a.
40 X8 Y6
5 XY3
8X7Y3
b.
X3 –2X2 –33X –7
X –7
X2 – 9X + 30 + -217
X–7
5. Simplify each expression:
a. ( -10 )0 1
b. 400X0 1
6. Write each number in scientific notation:
a. 73, 900, 000 7.39 x 107
b. 0.38 3.8 x 10-1
7. Factor by grouping:
a. X3 + 3X2 + 2X + 6 X2(X+3) + 2(X+3) = (X2+2)(X+3)
b. XY + 4X – 2Y – 8 X(Y+4) – 2(Y + 4) = (X – 2)(Y+4)
8.
Factor completely:
a. X2 –X –20 (X – 5)(X+4)
b. 12y3 + 28y2 + 8y y(12y2+28+8) = y(2y+4)(6y+2)
Factor completely or state that it is “Prime”:
9.
a. Z4 –16 (Z2+4)(Z2-4)
b. X2 + 1 prime
10. Factor Completely:
a. X2 –16X + 64 (X-8)2
b. X3 – 27 (X-3)(X2 + 3X + 9)
11. Solve each equation:
a. X(X –12 ) = 0 X=0, 12
b. 5X2 +20X = 0 5X(X+4)=0; X=0, -4
12. The length of a rectangular sign is 3feet longer than the width. If the sign has a
area of 40 square feet for advertising, find its length and its width.
L=3+w; Lw=40; (3+w)w=40; 3w+w2 = 40; w2+3w-40=0; (w+8)(w-5) = 0; w=-8, 5;
negative width impossible; w=5, L=8
13. Find all the numbers for which the rational expression is undefined:
a.
X+3
(X –2 )(X + 5)
X=2, -5
b.
7
X2 + 81
Expression defined for all real numbers, undefined for 9i
14. Simplify
a. X2 – 4
X – 2 X=2
b. y2 + 2y
y2 + 4y + 4 (y+2)2; y=-2
15.
Multiply or divide as indicated:
16.
a.
5X + 5 multiplied by
6
b.
X2 + X
10
3X
X2 + X
divided by 2X + 4
5
15X2+15X = 15(X2+X) = 5
6X2+6X
6(X2+X) 2
5X2+5X = 5(X2+X) = X2+X
20X+40 5(4X+8) 4X+8
Add or Subtract as indicated, Simplify the results, if possible:
a. 7
(X + 3)
b.
17.
6y
y2 - 4
+
4
7(X+3) + 4 = 7X+21+4 = (7X+25)
(X + 3)2 (X+3)2
(X+3)2
(X+3)2
-
3
y+2
6y – 3(y-2) = 6y – 6y+2 = 2
y2 – 4
y2 – 4
y2 - 4
Solve each irrational equation. If an equation has no solution, so state:
a. 3
X
-1 = 1
6
X
18-6x = 6
b. 13
- 3 = 1
y–1
y–1
18.
-6x = -12
13 – 3(y – 1) = 1
-3y = -15
y=5
x=2
13 – 3y+3 = 1
16 – 3y = 1
A car travels 60 miles in the same time that a car traveling 10 per hour faster
travels 90 miles. What is the rate of each cars?
60 = 90
x x+10
19.
60x+600 = 90x 600 = 30x
x = 20 20mph, 30 mph
Find the indicated root or state that the expression is not a real number:
a.
√8
125
3
2
5
This is the cube root of 8/125.
b. -4√81 = 81-0.25 = 1/3
20. Simplify each expression:
a. √45X23 √9*5X22X = 3X11√5X
b.
√ 96
3
√16*6 = 4√6
3
3
21. Add or subtract as indicated:
a. 4√72 - 2√48 4√2*2*2*3*3 – 2√2*2*2*2*3 = 24√2 – 8√3
b. √75 - √48 √25*3 - √16*3 = 5√3 – 4√3 = √3
22. Multiply as indicated:
a. (√11 - √ 5 ) (√11 +√ 5) 11 – 5 = 6
b. ( 1 + √2) 2 1 + 2√2 + 2
23.
Solve each radical expression:
a. √ (2X – 1) = X – 2
This the square root of what is in the parenthesis.
2X – 1 = X2 – 2X + 2
X2 – 4X + 3 = 0 (X-3)(X-1)
X= 3, 1
b. √3X + 5 = 11 This is the square root of 3x
√3X = 6
3X = 36
X=12
24. Simplify:
a. 64 2/3 3√642 = 42 = 16
b. (-8 ) -4/3 3√-8-4 1/3√84 = 3√4096 = 16
25. Solve by completing the square:
X2 –12X + 27 = 0
X-6 = 3, -3
X2 – 12X = -27 X2 – 12X + 36 = 9
X = 9, 3
(X-6)2 = 9
26. Solve by using the quadratic formula:
X2 = 2X + 4
a=1, b=-2, c=-3
D = 16 (-2+16)÷2=7
(-2-16)÷2=-9
X=7. -9
27. Solve each equation:
a. X2 = -100 X=10i or undefined
b. 5X2 = -125 X = 5i or undefined
Given the parabola y = x2 – 6x –7 , answer the following questions:
28. Determine if the parabola whose equation is given opens upward or downward.
What is its x-intercept? Upward; 7, 0; -1, 0
29. Find its y-intercept. What are the coordinates of the vertex? 0, -7; 3, -16
30. Graph the parabola. Graph pointing upward with vertex 3, -16 crossing the x axis
at x = 7 and –1 and crossing the y axis at y = -7