Math 105 Exam II Solution

1. Simplify . (10)
Solution :

2. Divide. (10)
(3x⁴ - 5x³ + 4x² - 5x + 1) (x² + 1)
Solution :
We cannot use synthetic division since the divisor is not of the form x-a. We use long
division instead after rewriting the divisor as x² + 0x + 1. We obtain 3x² - 5x + 1.

3. Use the remainder theorem to find f(4), where f(x) = x⁴ -x³ - 19x² + 49x - 30. (10)
Solution:
Use 4 for the synthetic division .

So f(4) = 54.

4. Solve the equation for p (10)

5. Simplify. (10)

Since the root is odd , we need no absolute values.

6. Divide and simplify . (10)

Since both x and y appear with odd powers under the radical in the original ex-
pression , they both had to be positive in the first place. Therefore, the final expression
needs no absolute value.

7. Factor completely .(10)
x²(x + 3) - 4(x + 3) = (x + 3)(x² - 4) = (x + 3)(x + 2)(x - 2)

8. Determine the domain of f. (10)

The domain of f is the set of values x ∈ R for which the denominator x²-7x+6 is not
equal to 0. To obtain the values that must be excluded from R, set the denominator
equal to zero :

The domain of f is therefore {x ∈ R l x≠ 1, x ≠ 6}.

9. Divide and, if possible, simplify. (10)
Solution:

10. Find the LCD , then add and simplify. (10)
Solution:

11. Rosanna walks 2 mph slower than Simone. In the time it takes Simone to walk 8 mi,
Rosanna walks 5 mi. Find the speed of each person. (10)
Solution:
 

Since d = r . t, we have t
and .Setting the equations equal we get

Simone walks at a rate ofmph and Rosanna at a rate ofmph.