Question

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

role="math" localid="1648185076360" $\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}$

Short answer

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

Step-by-step solution

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.