Thread: Need help on factor theorem.

I am supposed to use the Factor Theorem to either factor the polynomial completely or to prove that it has no linear factors with integer coefficients.
I think I haven't completely understood the whole concept yet, which is why I couldn't get my brain working on the following problems.

a. P(x)= x^4+4x^3-7x^2-34x-24
b. P(x)= x^6-6x^5+15x^4-20x^3+15x^2-6x+1

I think these are harder than the regular ones like P(x)=x^3+2x^2-9x+3, etc.

Thanks.

Hello zxazsw

Originally Posted by zxazsw

I am supposed to use the Factor Theorem to either factor the polynomial completely or to prove that it has no linear factors with integer coefficients.
I think I haven't completely understood the whole concept yet, which is why I couldn't get my brain working on the following problems.

a. P(x)= x^4+4x^3-7x^2-34x-24
b. P(x)= x^6-6x^5+15x^4-20x^3+15x^2-6x+1

I think these are harder than the regular ones like P(x)=x^3+2x^2-9x+3, etc.

Thanks.

Two things you need to use here:

• The Factor Theorem itself: is a factor of if and only if .

• If the constant term in is , then in any factor of the form , . must be a factor of

So in part (a), you'll only need to try values of which are factors of (not forgetting to try negative numbers as well), to see whether . If my arithmetic is correct, none of them works, so there aren't any linear factors with integer coefficients.

By the same reasoning, in part (b), are the only possibilities. So try first. And, even when you have found the first factor, keep on trying with the quotient until you're sure there are no factors left. (Hint: do you know anything about Pascal's triangle or the Binomial Theorem?)

Grandad