Selected Answers to Math 390 Review Exam 3
I. Polynomials: Perform the indicated operations.
A. 1. (3x² - 5x + 4) + (7x² + 9x - 8) = 3x² - 5x + 4 + 7x² + 9x - 8
= 10x² + 4x - 4
2. (7x² - 5x + 3) - (-2x² + 6x - 4) = 7x² - 5x + 3 + 2x² - 6x + 4
= 9x² - 11x + 7
3. 5x(x - 4) - 3x(2x + 1) = 5x² - 20x - 6x² - 3x = -x² - 23x
4. (6x² y)(4x³ y² ) = twenty four x to the fifth power y to the third power.
There is no way to indicate an exponent other than 2 or 3 hence the answer in words.
5. (3x - 8)(4x + 1) = 12x² + 3x - 32x - 8 = 12x² - 29x - 8
6. (2x² + 3x - 1)(x² + 5x + 4) = 2x to the fourth power + 10x³ + 8x² + 3x³ + 15x² + 12x - x² - 5x - 4
= 2x to the fourth power + 13³ + 22x² + 7x - 4
7. (4a³ )² = 16 a to the sixth power
8. (5x - 3)² = 25x² - 30x + 9
9. (35x³ y³ )/(7xy² ) = 5x² y
10. (14x³ - 22x)/(2x) = (14x³ )/(2x) - (22x)/2x = 7x² - 11
11. Answer: 2x² + 4x + 9 + 14/(x - 2)
B. 1. Subtract x² + 4x - 3 from 2x² + 7x - 8.
(2x² + 7x - 8) - (x² + 4x - 3) = 2x² + 7x - 8 - x² - 4x + 3
= x² + 3x - 5
2. What is the degree of the polynomial 4x² + 7x³ - 3x + 9 ?
The polynomial is of degree three.
II. Factoring.
A. Completely factor.
1. 3x² - 9x - 30 = 3(x² - 3x - 10) = 3(x - 5)(x + 2)
2. 6x² - 7x - 10 = (6x + 5 )(x - 2 )
3. 50x² - 32 = 2(25x² - 16) = 2(5x + 4)(5x - 4)
4. 49x² - 42x + 9 = (7x - 3)² Did you recognize the fact the trinomial was a perfect square trinomial?
5. x² - 5x - 36 = (x - 9)(x + 4)
6. 125x³ - 64 = (5x - 4)(25x² + 20x + 16)
7. 27x³ + 8 = (3x + 2)(9x² - 6x + 4)
B. Solve for x.
1. 2x² - 3 = -5x
2x² + 5x - 3 = 0
(2x - 1)(x + 3) = 0
Either 2x - 1 = 0 or x + 3 = 0
Either 2x = 1 or x = -3
Either x = 1/2 or x = -3
2. 4x² - 20x + 25 = 0
(2x - 5)² = 0
2x - 5 = 0
2x = 5
x = 5/2
3. 5x² + 6x - 8 = 0
(5x - 4)(x + 2) = 0
Either 5x - 4 = 0 or x + 2 = 0
Either 5x = 4 or x = -2
Either x = 4/5 or x = -2
C. Stated Problems.
1. If three times the square of a certain integer is added to the integer itself,
the sum is 10. What is the integer?
Let x denote the integer.
3x² + x = 10
3x&$178 + x - 10 = 0
(3x - 5)(x + 2) = 0
Either 3x - 5 = 0 or x + 2 = 0
3x = 5 or x = -2
x = 5/3 or x = -2
Note 5/3 is not an integer. Hence the integer is -2.
2. The base of a triangle is four feet shorter than twice its altitude. If the area
of the triangle is 48 square feet, find its dimensions.
Let x denote the altitude. Then 2x - 4 denotes the base.
Remeber the area A of a triangle is given by the formula A = 1/2(bh)
48 = 1/2(x)(2x - 4)
96 = x(2x - 4)
96 = 2x² - 4x
0 = 2x² - 4x - 96
0 = 2(x² - 2x - 48)
0 = 2(x - 8)(x + 6)
Either x - 8 = 0 or x + 6 = 0
Either x = 8 or x = -6
Since the altitude can't be negative the altitude is 8, the base is 2x - 4 or 2(8) - 4 = 12.
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