# 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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