To factor out the greatest common factor (GCF) from the polynomial [tex]\(30y^6 + 5y^2\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- The coefficients in the polynomial are 30 and 5.
- The greatest common factor of 30 and 5 is 5.
2. Identify the lowest power of [tex]\( y \)[/tex]:
- The exponents of [tex]\( y \)[/tex] in the terms are 6 and 2.
- The lowest power of [tex]\( y \)[/tex] is [tex]\( y^2 \)[/tex].
3. Factor out the GCF:
- Factor 5 and [tex]\( y^2 \)[/tex] out of each term in the polynomial.
So, let's write down the original polynomial and factor out [tex]\( 5y^2 \)[/tex]:
[tex]\[ 30y^6 + 5y^2 \][/tex]
Factoring out [tex]\( 5y^2 \)[/tex]:
[tex]\[ 30y^6 + 5y^2 = 5y^2(6y^4 + 1) \][/tex]
4. Simplify the expression inside the parentheses to confirm it’s correct:
- Distribute [tex]\( 5y^2 \)[/tex] back into [tex]\( 6y^4 + 1 \)[/tex] to check:
[tex]\[ 5y^2(6y^4 + 1) = 5y^2 \cdot 6y^4 + 5y^2 \cdot 1 = 30y^6 + 5y^2 \][/tex]
So, the polynomial [tex]\(30y^6 + 5y^2\)[/tex] factored with the GCF is:
[tex]\[ 30y^6 + 5y^2 = 5y^2 (6y^4 + 1) \][/tex]
This is the simplified answer.
