SOLUTION: Please, I need help with this problem: Assume that (X-2) is a factor of the polynomial f(x)=x^3+a(x^2)+bx+2 and that f(x) gives a remainder of -3 when it is divided by x+1. Then,

Question 31083: Please, I need help with this problem:
Assume that (X-2) is a factor of the polynomial f(x)=x^3+a(x^2)+bx+2 and that f(x) gives a remainder of -3 when it is divided by x+1. Then, a=?, b=?.
What I did is I used the factor they gave me and used the 2 as a root. It gave me 4a+2b=-10 and so I planned to use and solve a system of equations with what I could get from the other information they give me with the problem. I'm stuck though, cause I have no idea of what to do with the remainder data...
Please help, and thanks so much!
Answer by mukhopadhyay(490) (Show Source): You can put this solution on YOUR website!
f(x)=x^3+a(x^2)+bx+2
Since (x-2) is a factor, f(2) should be zero
f(2)=2^3+a(2^2)+2b+2=0
=>8 + 4a + 2b + 2=0
=>4a + 2b + 10 = 0
=>2(2a + b + 5) = 0
=>2a + b = -5
When divided by (x+1), the remainder is -3
=>f(-1)=-3
=>(-1)^3 + a(-1)^2 + 2b + 2 = -3
=>-1 + a + 2b + 2 = -3
=>a + 2b + 1 = -3
=>a + 2b = -4
2a + b = -5 =>b = -5 - 2a
Substituting b in the other equation a + 2b = -4 gives
a + 2(-5 - 2a) = -4
=>a - 4a = 6
=>-3a=6
=>a=-2
So, b = -5 - 2(-2) = -1
Answer: a=-2 and b=-1

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