when a polynomial f(x)= x^3+ax^2-20x+24 is divided by x-1, the remainder is 7. Find the value of a. With the value of a, find the three linear factor s of f(x).

Question

anonymous · Answer

i found the value of a which is 2 , but i cant get the the answer correctly for the following question.

hero · Answer

Interesting. How'd you figure out that the a = 2? Here's what I did:

If x^3 + ax^2 - 20x + 24 is divided by x - 1, then it has remainder 7. Therefore
x^3 + ax^2 -20x + 17 should divide evenly with x - 1

So we need to split ax^2 so that x - 1 divides evenly:

x^3 + ax^2 - 20x + 17 = x^3 - x^2 + 3x^2 - 3x -17x + 17
                                        = x^2(x - 1) + 3x(x - 1) - 17(x - 1)
                                        = (x - 1)(x^2 + 3x - 17)

So obviously -x^2 + 3x^2 = 2x^2 which means a = 2

hero · Answer

Now that we know a = 2, we replace a with 2 in the original polynomial:

f(x)= x^3+2x^2-20x+24

Then factor it. By factor theorem, f(2) = 0, which means x - 2 is a factor. Therefore:

x^3 + 2x^2 - 20x + 24 = x^3 - 2x^2 + 4x^2 - 8x -12x + 24
                                    = x^2(x - 2) + 4x(x - 2) - 12(x - 2)
                                    = (x - 2)(x^2 + 4x - 12)

Finish factoring x^2 + 4x - 12 to find the remaining factors.

anonymous · Answer

the way i found a=2 is since x-1 is a factor of f(x) so
f(1)=(1)^3+a(1)-20(1)+24
7=1+a-20+24
a=2

hero · Answer

Yep, that's the easiest way to find it.

anonymous · Answer

oh okay ,thanks for your help , but actually i am in trouble with this type of question , might have more practices on it.