# Math 105 Exam II Solution

2. Divide. (10)

(3x^{4} - 5x^{3} + 4x^{2} - 5x + 1)
(x^{2} + 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 x^{2} + 0x + 1. We
obtain 3x^{2} - 5x + 1.

3. Use the **remainder theorem** to find f(4), where f(x) = x^{4} -x^{3}
- 19x^{2} + 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)

x^{2}(x + 3) - 4(x + 3) = (x + 3)(x^{2} - 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 x^{2}-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:

d | r | t | |

Rosanna | 5 | x - 2 | t |

Simone | 8 | x | t |

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.

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