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

Answer:

$a = -6, b = - 8$

Remaining factor = $x+4$

Step-by-step 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 $x = 2$:

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

Putting $x = -1$:

$(-1)^3 +3 \times (-1)^2 +a\times (-1) +b =0\\\Rightarrow -a+b=-2 ..... (2)$

Subtracting (2) from (1):

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

From equation (1):

$2\times (-6) + b=-20\\\Rightarrow b = -8$

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:

$\dfrac{x^3 +3 x^2 -6x -8}{x^{2} -x-2} = x+4$