Question

Given that x² -x -2 is a factor of x³ +3 x² +ax +b, calculate the values of a and b and hence find the remaining factor.

Answer 1

`a = -6, b = - 8`

Remaining factor =
`x+4`

Explanation:

Given that:

Two polynomials:

`x^(2) -x-2` and
`x^3 +3 x^2 +ax +b`

`x^(2) -x-2` is a factor of
`x^3 +3 x^2 +ax +b`.

To find:

The values of
`a` and
`b` from the cubic equation and the remaining factor.

Solution:

Let us first of all, factorize
`x^(2) -x-2`.

`x^(2) -2x+x-2=0\\\Rightarrow x(x-2)+1(x-2)=0\\\Rightarrow (x+1)(x-2)=0\\\Rightarrow x = -1, 2`

There are two factors of the given quadratic equation.

These two factors must also be factors of the cubic equation as well.

Putting
`x = 2`:

`2^3 +3 * 2^2 +a* 2 +b =0\\\Rightarrow 2a+b=-20 ..... (1)`

Putting
`x = -1`:

`(-1)^3 +3 * (-1)^2 +a* (-1) +b =0\\\Rightarrow -a+b=-2 ..... (2)`

Subtracting (2) from (1):

`3a = -18\\\Rightarrow a = -6`

From equation (1):

`2* (-6) + b=-20\\\Rightarrow b = -8`

Putting the values of
`a` and
`b` in the cubic equation, we get:

`x^3 +3 x^2 -6x -8`

Dividing the cubic equation with quadratic, we get:

`(x^3 +3 x^2 -6x -8)/(x^(2) -x-2) = x+4`