Question

Factor.
4 x Superscript 5 Baseline minus 324 x

Answer 1

To factor the expression \( 4x^5 - 324x \), we can follow these steps:

1. **Factor out the greatest common factor (GCF)**:
The GCF of \( 4x^5 \) and \( 324x \) is \( 4x \).

So we can factor out \( 4x \):
\[
4x(x^4 - 81)
\]

2. **Recognize and factor the difference of squares**:
Notice that \( x^4 - 81 \) is a difference of squares since \( 81 = 9^2 \):
\[
x^4 - 9^2 = (x^2 - 9)(x^2 + 9)
\]

3. **Factor \( x^2 - 9 \) further**:
\( x^2 - 9 \) is also a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

Putting all of this together, we have:
\[
4x(x^4 - 81) = 4x(x^2 - 9)(x^2 + 9) = 4x(x - 3)(x + 3)(x^2 + 9)
\]

Thus, the fully factored form of the expression \( 4x^5 - 324x \) is:
\[
\boxed{4x(x - 3)(x + 3)(x^2 + 9)}
\]