Question

Factor the expression completely.

$x^2\left(x^2-1\right)-9\left(x^2-1\right)$

Short answer

The solution$x^2\left(x^2-1\right)-9\left(x^2-1\right)=\left(x-3\right)\left(x+3\right)\left(x-1\right)\left(x+1\right)$

Step-by-step solution

Step 1. Apply the Factoring Formula.

If A and B are any real numbers or algebraic expressions, then

$A^2-B^2=\left(A+B\right)\left(A-B\right)$

Step 2. Analyze the given expression.

Given$x^2\left(x^2-1\right)-9\left(x^2-1\right)$

First using distributive property, we write the expression as:

$x^2\left(x^2-1\right)-9\left(x^2-1\right)=\left(x^2-9\right)\left(x^2-1\right)$

Step 3. Solve the expression.

Substitute the values in the Factoring formula to get:

$\begin{array}{l}{x}^2\left(x^2-1\right)-9\left(x^2-1\right)=\left(x^2-9\right)\left(x^2-1\right)\\ \Rightarrow x^2\left(x^2-1\right)-9\left(x^2-1\right)=\left(x^2-3^2\right)\left(x^2-1^2\right)\\ \Rightarrow x^2\left(x^2-1\right)-9\left(x^2-1\right)=\left(x-3\right)\left(x+3\right)\left(x-1\right)\left(x+1\right)\end{array}$