To factor the greatest common factor (GCF) out of the expression [tex]\( 72x^6 + 40x^4 \)[/tex], follow these steps:
1. Identify the coefficients and variables: The coefficients are 72 and 40, and the corresponding exponents of [tex]\(x\)[/tex] are 6 and 4.
2. Find the GCF of the coefficients:
- The greatest common factor of 72 and 40 is 8. This is because the largest number that can divide both 72 and 40 evenly is 8.
3. Determine the GCF of the variable terms:
- For the exponents of [tex]\(x\)[/tex], the GCF is determined by the lowest exponent among them. The exponents are 6 and 4, so the lower of these values is 4.
- Therefore, the GCF of [tex]\(x^6\)[/tex] and [tex]\(x^4\)[/tex] is [tex]\(x^4\)[/tex].
4. Combine the GCF of the coefficients with the GCF of the variables:
- The GCF of the entire expression is [tex]\(8x^4\)[/tex].
5. Factor out the GCF:
- To do this, divide each term in the original expression by the GCF [tex]\(8x^4\)[/tex]:
[tex]\[
\frac{72x^6}{8x^4} = 9x^2
\][/tex]
[tex]\[
\frac{40x^4}{8x^4} = 5
\][/tex]
6. Write the factored form:
- After factoring out the GCF, the expression becomes:
[tex]\[
72x^6 + 40x^4 = 8x^4(9x^2 + 5)
\][/tex]
Therefore, the expression [tex]\( 72x^6 + 40x^4 \)[/tex] factored with the GCF is:
[tex]\[
8x^4(9x^2 + 5)
\][/tex]
In this form, [tex]\(A = 8x^4\)[/tex], [tex]\(B = 9x^2\)[/tex], and [tex]\(C = 5\)[/tex], so the expression is written as [tex]\(A(B + C)\)[/tex].
