To factor the polynomial [tex]\( 12 c^9 + 28 c^7 \)[/tex], let's follow the steps one by one:
1. Find the Greatest Common Factor (GCF):
- For the coefficients 12 and 28, the GCF is 4.
- For the variable parts [tex]\( c^9 \)[/tex] and [tex]\( c^7 \)[/tex], the GCF is [tex]\( c^7 \)[/tex] since it's the smallest power of [tex]\( c \)[/tex] common to both terms.
Therefore, the GCF of [tex]\( 12 c^9 \)[/tex] and [tex]\( 28 c^7 \)[/tex] is [tex]\( 4 c^7 \)[/tex].
[tex]\[
\text{GCF} = 4 c^7
\][/tex]
2. Write each term as a product, where one factor is the GCF:
- For [tex]\( 12 c^9 \)[/tex]:
[tex]\[
12 c^9 = (4 c^7) \cdot 3 c^2
\][/tex]
- For [tex]\( 28 c^7 \)[/tex]:
[tex]\[
28 c^7 = (4 c^7) \cdot 7
\][/tex]
So, we can rewrite the polynomial as:
[tex]\[
12 c^9 + 28 c^7 = 4 c^7 (3 c^2) + 4 c^7 (7)
\][/tex]
3. Use the distributive property:
Factor out the common term [tex]\( 4 c^7 \)[/tex]:
[tex]\[
12 c^9 + 28 c^7 = 4 c^7 (3 c^2 + 7)
\][/tex]
The correct resulting expression is:
[tex]\[
4 c^7 (3 c^2 + 7)
\][/tex]
So, the correct answer is:
[tex]\[
4 c^7 \left( 3 c^2 + 7 \right)
\][/tex]
