Certainly! Let's factor the expression [tex]\(100 x^4 - 30 x^2\)[/tex] completely step-by-step.
1. Identify the Greatest Common Factor (GCF):
First, notice that both terms in the expression [tex]\(100 x^4 - 30 x^2\)[/tex] have a common factor. The greatest common factor (GCF) of the numerical coefficients 100 and 30 is 10. Additionally, both terms include [tex]\(x^2\)[/tex].
So, the GCF of the entire expression is [tex]\(10 x^2\)[/tex].
2. Factor out the GCF:
We can factor out [tex]\(10 x^2\)[/tex] from each term:
[tex]\[
100 x^4 - 30 x^2 = 10 x^2 (10 x^2) - 10 x^2 (3)
\][/tex]
When we factor out [tex]\(10 x^2\)[/tex], the expression becomes:
[tex]\[
100 x^4 - 30 x^2 = 10 x^2 (10 x^2 - 3)
\][/tex]
3. Check for Further Factoring:
Now, examine the remaining expression inside the parentheses, [tex]\(10 x^2 - 3\)[/tex], to see if it can be factored further.
The expression [tex]\(10 x^2 - 3\)[/tex] does not factor any further within the real numbers because it is a quadratic expression in standard form and does not have rational factors.
Thus, the completely factored form of [tex]\(100 x^4 - 30 x^2\)[/tex] is:
[tex]\[
10 x^2 (10 x^2 - 3)
\][/tex]
So, the final factored expression is:
[tex]\[
\boxed{10 x^2 (10 x^2 - 3)}
\][/tex]
