SOLUTION: 1.) Explain the difference between using the remainder theorem and synthetic division in finding the remainder when P(x) is divided by x-r.

I think you're asking what is the difference between the 
remainder theorem and the factor theorem?

The Remainder Theorem:

When we divide a polynomial f(x) by x-c the remainder equals f(c)

The Factor Theorem:

When f(c)=0 then x-c is a factor of the polynomial
When x-c is a factor of the polynomial then f(c)=0

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1.  Use the remainder theorem to find what you get when you substitute
    -3 for x in f(x) = 3x^3 - 5x^2 + 7x - 4, whch is f(-3).  Then check 
    your answer by actually substituting -3 in f(x) to find f(-3)

    -3 | 3   -5    7    -4
       |     -9   42  -147
         3  -14   49  -151

So f(-3) = -151, the number in the lower right corner of the synthetic division. 

Checking by actually substituting -3 for x in f(x)

f(x) = 3x^3 - 5x^2 + 7x - 4
f(-3) = 3(-3)^3 - 5(-3)^2 + 7(-3) - 4
f(-3) = 3(-27) - 5(9) -21 - 4
f(-3) = -81 - 45 - 21 - 4
f(-3) = -151


2.  Use the factor theorem to show that x-2 is a factor of x^4-7x^2+x+10

2 | 1  0 -7  1  10
  |    2  4 -6 -10 
    1  2 -3 -5   0

Edwin