Q3. - Algebra 1

Question

Which of the following is not a factor of ${\mathit{x}}^4\mathbf{-}\mathbf{6}{\mathit{x}}^2\mathbf{-}\mathbf{27}$?

$\mathit{A}$ ${\mathit{x}}^2\mathbf{+}\mathbf{3}$    $\mathit{C}$ $\mathit{x}\mathbf{+}\mathbf{3}$

$\mathit{B}$ $\mathit{x}\mathbf{-}\mathbf{3}$     $\mathit{D}$ ${\mathit{x}}^2\mathbf{-}\mathbf{3}$

Step-by-Step Solution

Answer

${\mathbf{x}}^2\mathbf{-}\mathbf{3}$ is not a factor of the given expression.

Step 1. Substitution.

Substitute $z$ for $x^2$ into the given expression.

 $z^2-6z-27$

Step 2. Factor.

Factor the above obtained expression.

 $\begin{array}{l}{z}^2+3z-9z-27=z(z+3)-9(z+3)\\ =(z-9)(z+3)\end{array}$

Step 3. Substitution.

Substitute $x^2$ for $z$ into $(z-9)(z+3)$.

 $\left(x^2-9\right)\left(x^2+3\right)=\left(x+3\right)\left(x-3\right)\left(x^2+3\right)$ 

Therefore, $x^2-3$ is not a factor of given expression. Option $D$ is the required answer.