find the values for a and b for which x^3+ax^2-10x+b is exactly divisible by x^2+x-12, and then factorise the cubic.

Answer 1

To find the values for a and b for which x^3 + ax^2 - 10x + b is exactly divisible by x^2 + x - 12, we need to perform polynomial long division.

Dividing x^3 + ax^2 - 10x + b by x^2 + x - 12, we get:

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x^2 + x - 12 | x^3 + ax^2 - 10x + b
- (x^3 + x^2 - 12x)
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ax^2 + 2x + b

For x^3 + ax^2 - 10x + b to be exactly divisible by x^2 + x - 12, the remainder ax^2 + 2x + b should be equal to 0. This means that:

a = 0
2 = -10
b = -12

Therefore, the values for a and b that make the polynomial divisible are a = 0 and b = -12.

Now, the cubic x^3 + ax^2 - 10x + b simplifies to x^3 -10x -12.

To factorise the cubic x^3 -10x -12, we can look for rational roots. By testing values using the Rational Root Theorem, we find that x = 2 is a root of the cubic.

Therefore, x^3 -10x -12 can be factorized as (x-2)(x^2 + 2x + 6).

Therefore, the values for a and b are a = 0 and b = -12, and the factorised form of the cubic is (x-2)(x^2 + 2x + 6).