To factor the greatest common factor (GCF) out of the expression [tex]\( 72x^6 + 40x^4 \)[/tex], follow these steps:
### Step 1: Identify the greatest common factor (GCF)
First, we need to find the GCF of the coefficients [tex]\( 72 \)[/tex] and [tex]\( 40 \)[/tex].
- The prime factorization of [tex]\( 72 \)[/tex] is [tex]\( 2^3 \times 3^2 \)[/tex].
- The prime factorization of [tex]\( 40 \)[/tex] is [tex]\( 2^3 \times 5 \)[/tex].
The highest power of the common prime factors is [tex]\( 2^3 = 8 \)[/tex].
Thus, the GCF of [tex]\( 72 \)[/tex] and [tex]\( 40 \)[/tex] is [tex]\( 8 \)[/tex].
Next, we consider the variables [tex]\( x^6 \)[/tex] and [tex]\( x^4 \)[/tex].
- The highest power of [tex]\( x \)[/tex] that is common in both terms is [tex]\( x^4 \)[/tex].
Therefore, the GCF of the expression [tex]\( 72x^6 + 40x^4 \)[/tex] is [tex]\( 8x^4 \)[/tex].
### Step 2: Factor out the GCF from each term
Now, we factor [tex]\( 8x^4 \)[/tex] out of [tex]\( 72x^6 \)[/tex] and [tex]\( 40x^4 \)[/tex]:
[tex]\[
72x^6 \div 8x^4 = 9x^2
\][/tex]
[tex]\[
40x^4 \div 8x^4 = 5
\][/tex]
### Step 3: Write the factored form of the expression
Thus, factoring [tex]\( 8x^4 \)[/tex] out of [tex]\( 72x^6 + 40x^4 \)[/tex] gives us:
[tex]\[
8x^4 (9x^2 + 5)
\][/tex]
### Final Answer:
The expression [tex]\( 72x^6 + 40x^4 \)[/tex] factored by its greatest common factor (GCF) is [tex]\( 8x^4 (9x^2 + 5) \)[/tex].
