Answer :
We start with the expression
[tex]$$
x^5 - 25x^3.
$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
Both terms in the expression have a common factor of [tex]$x^3$[/tex], so we factor that out:
[tex]$$
x^5 - 25x^3 = x^3 \left(x^2 - 25\right).
$$[/tex]
Step 2. Factor the Remaining Expression:
The expression inside the parentheses,
[tex]$$
x^2 - 25,
$$[/tex]
is a difference of squares, which factors according to the formula
[tex]$$
a^2 - b^2 = (a + b)(a - b).
$$[/tex]
Here, [tex]$a = x$[/tex] and [tex]$b = 5$[/tex], so
[tex]$$
x^2 - 25 = (x + 5)(x - 5).
$$[/tex]
Step 3. Write the Fully Factored Form:
Substitute the factored form of [tex]$x^2 - 25$[/tex] back into the expression:
[tex]$$
x^5 - 25x^3 = x^3 \cdot (x + 5)(x - 5).
$$[/tex]
Thus, the final factored form is
[tex]$$
\boxed{x^3 (x - 5)(x + 5)}.
$$[/tex]
[tex]$$
x^5 - 25x^3.
$$[/tex]
Step 1. Factor out the Greatest Common Factor (GCF):
Both terms in the expression have a common factor of [tex]$x^3$[/tex], so we factor that out:
[tex]$$
x^5 - 25x^3 = x^3 \left(x^2 - 25\right).
$$[/tex]
Step 2. Factor the Remaining Expression:
The expression inside the parentheses,
[tex]$$
x^2 - 25,
$$[/tex]
is a difference of squares, which factors according to the formula
[tex]$$
a^2 - b^2 = (a + b)(a - b).
$$[/tex]
Here, [tex]$a = x$[/tex] and [tex]$b = 5$[/tex], so
[tex]$$
x^2 - 25 = (x + 5)(x - 5).
$$[/tex]
Step 3. Write the Fully Factored Form:
Substitute the factored form of [tex]$x^2 - 25$[/tex] back into the expression:
[tex]$$
x^5 - 25x^3 = x^3 \cdot (x + 5)(x - 5).
$$[/tex]
Thus, the final factored form is
[tex]$$
\boxed{x^3 (x - 5)(x + 5)}.
$$[/tex]
