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**RickyOswaldIOW****Member**- Registered: 2005-11-18
- Posts: 212

I have an expression: f(x) = x^3 + ax^2 + bx + c which is divisable by x. Does this mean that

f(0) = (0)^3 + a(0)^2 + b(0) + c = 0 and thus c = 0?

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,470

The factor theorem states that a polynomial f(x) is divisible by (x-a) if f(a)=0.

The given polynomial is

Since it is divisible by x,

it follows that f(x)=0, therefore, the given polynomial is equal to zero, and c=0. True, because, if c is a non-zero constant, the polynomial is not divisible by x.

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**RickyOswaldIOW****Member**- Registered: 2005-11-18
- Posts: 212

so if f(3) = 0 and f(-2) = -40, what are the values of a and b? I've been bearing in mind that c = 0 to work this out but I always end up with fractions in my answer, the book says a and b are integers.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,470

Since f(3)=0, 27+9a+3b+c=0,...(1)

f(-2)=-40, therefore, -8+4a-2b+c=-40....(2)

Since c=0,

9a+3b=-27...(3)

4a-2b=-32...(4)

18a+6b=-54

12a-6b=-96

30a=-150, a=-5, b=6.

***I think....I haven't double checked my answer****

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**RickyOswaldIOW****Member**- Registered: 2005-11-18
- Posts: 212

I did this question 3 times seperately, each time I made the same mistake. (3)^3 = 27 as you state in the first line, I put down 9 each time!!!

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