Question

Fully factor the following high degree polynomial.

X^4 - 25x^2

A. x(x + 10) (x-10)
B. x^2 (x+5) (x-5)
C. (x^4 + 5x^2) (x^3 - 5)
D. (x^3 + 10x^2) (x^4 - 10x)

Answer 1

To factor the polynomial \( X^4 - 25x^2 \), we can start by factoring out the common term, which in this case is \( x^2 \).

1. **Factoring out \( x^2 \)**:
\[
x^2(X^2 - 25)
\]

2. **Recognizing the difference of squares**:
The term \( X^2 - 25 \) can be factored as a difference of squares:
\[
X^2 - 25 = (X - 5)(X + 5)
\]

3. **Putting it all together**:
Therefore, we can write the complete factorization:
\[
X^4 - 25x^2 = x^2 (X - 5)(X + 5)
\]

So, the fully factored form of the polynomial \( X^4 - 25x^2 \) is:
\[
x^2 (x - 5)(x + 5)
\]

Now, let's compare this with the provided options:

A. \( x(x + 10)(x - 10) \)

B. \( x^2 (x + 5)(x - 5) \)

C. \( (x^4 + 5x^2)(x^3 - 5) \)

D. \( (x^3 + 10x^2)(x^4 - 10x) \)

The correct option that matches our factorization \( x^2 (x - 5)(x + 5) \) is:

**B. \( x^2 (x + 5)(x - 5) \)**.