A little help with this Pre-Calculus problem???

Question

Determine a and b so that when (x^4)+(x^3)-(7x^2)+ax+b is divided by (x-1)(x+2), the remainder is 0. 10 points will be awarded to the answer with the most detail and accuracy. Thank you.

Answer 1

Synthetic division (I'm assuming you know how to do it....):

+01 | +01 +01 −07  + a + b
       | +00 +01 +02 −05 a-5
       |- - - - - - - - - - - - - - - -
       | +01 +02 −05 a-5 +00

You are given that (x - 1) must be a factor. That means that the sum of the last column must be 0. Therefore, we can see that it must be true that:

b + a - 5 = 0
b + a = 5

That's one equation.... Now we're also given that 2 must be a factor, so we divide the result of our last division:

x³ + 2x² -5x + (a - 5)

by -2.

−02 | +01 +02 −05 +a-5
       | +00 −02 +00 +10
       |- - - - - - - - - - - - - -
       | +01 +00 −05 +00

From the last column again, we can see that:

a - 5 + 10 = 0
a + 5 = 0
a = -5

Then from the first equation we compute b:

a + b = 5
-5 + b = 5
b = 10

So the original equation must have been:

x^4 + x³ - 7x² - 5x + 10

The easiest way to check to see if this works is just to plug in the roots 1 and -2 and verify that you get 0 for both.

1^4 + 1³ - 7(1)² - 5(1) + 10
= 1 + 1 - 7 - 5 + 10 = 0...check!

(-2)^4 + (-2)³ - 7(-2)² - 5(-2) + 10
= 16 - 8 - 28 + 10 + 10 = 0....check!

So I'm pretty sure this is right. a = -5 and b = 10.