Answer :

To factor the polynomial [tex]\(48 x^3 - 12 x^5\)[/tex], follow these steps:

1. Identify the Greatest Common Factor (GCF) of the coefficients.
- Look at the numbers 48 and 12.
- The GCF of 48 and 12 is 12.

2. Determine the lowest power of [tex]\(x\)[/tex] in the terms.
- We have [tex]\(x^3\)[/tex] and [tex]\(x^5\)[/tex].
- The lowest power of [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].

3. Factor out the GCF and the lowest power of [tex]\(x\)[/tex].
- From both terms, factor out 12 and [tex]\(x^3\)[/tex].
- This means we write [tex]\(48 x^3 - 12 x^5\)[/tex] as [tex]\(12 x^3 ( \frac{48}{12} \cdot x^{3-3} - \frac{12}{12} \cdot x^{5-3})\)[/tex].

4. Simplify inside the parentheses.
- [tex]\(48 \div 12 = 4\)[/tex], and [tex]\(x^{3-3} = x^0 = 1\)[/tex], so the first term becomes [tex]\(4\)[/tex].
- [tex]\(12 \div 12 = 1\)[/tex], and [tex]\(x^{5-3} = x^2\)[/tex], so the second term becomes [tex]\(x^2\)[/tex].

5. Write the final factored form.
- After factoring out [tex]\(12x^3\)[/tex], we get:
[tex]\[
12 x^3 (4 - x^2)
\][/tex]

So, the factored form of [tex]\(48 x^3 - 12 x^5\)[/tex] is:
[tex]\[
12 x^3 (4 - x^2)
\][/tex]

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