Polynomial problem solving

[Bethan]

Hi, I've been stuck on this one for what feels like an eternity. Any help would be appreciated!
I can solve for one missing term, but I cannot get my head around this one.

The polynomial 6x^3 +7x^2 +ax+b has a remainder of 72 when divided by (x-2) and is exactly divisible by (x+1). Find the values of a and b.

 

[MarkFL]

Hello, and welcome to FMH!

Let \(f(x)=6x^3+7x^2+ax+b\).

We are essentially told:

[MATH]f(x)=(x-2)Q(x)+72[/MATH]
Hence:

[MATH]f(2)=72[/MATH]
We also know:

[MATH]f(-1)=0[/MATH]
This will give you two equations in the two unknowns...can you proceed?

 

[Bethan]

Finally worked it out. I was overcomplicating it. Thank you

 

[MarkFL]

Yes good! I would have written:

[MATH]6\cdot2^3+7\cdot2^2+2a+b=72\implies 2a+b=-4[/MATH]
[MATH]6\cdot(-1)^3+7\cdot(-1)^2-a+b=0\implies a-b=1[/MATH]
Adding the two equation, we obtain:

[MATH]3a=-3\implies a=-1\implies b=-2[/MATH]