Answer :
To factor the expression [tex]\(2x^6 + 32x^4\)[/tex], we can follow these steps:
1. Identify the Greatest Common Factor (GCF):
- Look at both terms [tex]\(2x^6\)[/tex] and [tex]\(32x^4\)[/tex].
- The common factor for the coefficients 2 and 32 is 2.
- The common factor for the powers of [tex]\(x^6\)[/tex] and [tex]\(x^4\)[/tex] is [tex]\(x^4\)[/tex].
- Thus, the GCF of the expression is [tex]\(2x^4\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF [tex]\(2x^4\)[/tex]:
- [tex]\( \frac{2x^6}{2x^4} = x^2\)[/tex]
- [tex]\( \frac{32x^4}{2x^4} = 16\)[/tex]
- After factoring out [tex]\(2x^4\)[/tex], the expression becomes:
[tex]\[
2x^4(x^2 + 16)
\][/tex]
3. Check for Further Factoring:
- The expression inside the parentheses [tex]\(x^2 + 16\)[/tex] is a sum of squares, which typically cannot be factored further using real numbers. So, we leave it as is.
Therefore, the fully factored form of [tex]\(2x^6 + 32x^4\)[/tex] is:
[tex]\[
2x^4(x^2 + 16)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
- Look at both terms [tex]\(2x^6\)[/tex] and [tex]\(32x^4\)[/tex].
- The common factor for the coefficients 2 and 32 is 2.
- The common factor for the powers of [tex]\(x^6\)[/tex] and [tex]\(x^4\)[/tex] is [tex]\(x^4\)[/tex].
- Thus, the GCF of the expression is [tex]\(2x^4\)[/tex].
2. Factor out the GCF:
- Divide each term by the GCF [tex]\(2x^4\)[/tex]:
- [tex]\( \frac{2x^6}{2x^4} = x^2\)[/tex]
- [tex]\( \frac{32x^4}{2x^4} = 16\)[/tex]
- After factoring out [tex]\(2x^4\)[/tex], the expression becomes:
[tex]\[
2x^4(x^2 + 16)
\][/tex]
3. Check for Further Factoring:
- The expression inside the parentheses [tex]\(x^2 + 16\)[/tex] is a sum of squares, which typically cannot be factored further using real numbers. So, we leave it as is.
Therefore, the fully factored form of [tex]\(2x^6 + 32x^4\)[/tex] is:
[tex]\[
2x^4(x^2 + 16)
\][/tex]
