Factor the polynomial completely: x^4 - x^2
Understand the Problem
The question asks to factor the polynomial completely. The specific polynomial to factor is x^4 - x^2.
Answer
$x^2(x-1)(x+1)$
Answer for screen readers
$x^2(x-1)(x+1)$
Steps to Solve
- Identify the common factor
Both terms in the polynomial $x^4 - x^2$ have $x^2$ as a common factor.
- Factor out the common factor
Factor out $x^2$ from the polynomial: $$x^4 - x^2 = x^2(x^2 - 1)$$
- Recognize the difference of squares
The expression $(x^2 - 1)$ is a difference of squares, which can be factored as $(x-1)(x+1)$.
- Factor the difference of squares
Factor $(x^2 - 1)$ into $(x-1)(x+1)$: $$x^2(x^2 - 1) = x^2(x-1)(x+1)$$
- Final factored form
The polynomial is now completely factored.
$x^2(x-1)(x+1)$
More Information
The original polynomial was a quartic (degree 4) polynomial. After factoring, it is expressed as a product of a quadratic term ($x^2$) and two linear terms ($x-1$ and $x+1$).
Tips
A common mistake is to stop after factoring out $x^2$, leaving the answer as $x^2(x^2-1)$. It is important to recognize that $x^2-1$ can be further factored as a difference of squares.
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