Sure! Let's factor the polynomial [tex]\( 35s^6 + 7s^2 \)[/tex] completely by identifying and factoring out the greatest common factor (GCF). Here are the step-by-step instructions:
1. Identify the greatest common factor (GCF):
- Look at the coefficients: The GCF of 35 and 7 is 7.
- Look at the variable part: The polynomial involves [tex]\( s^6 \)[/tex] and [tex]\( s^2 \)[/tex]. The smallest power of [tex]\( s \)[/tex] common to both terms is [tex]\( s^2 \)[/tex].
Thus, the GCF of the entire polynomial is [tex]\( 7s^2 \)[/tex].
2. Factor out the GCF:
- Write each term as a product of [tex]\( 7s^2 \)[/tex] and another factor.
We can express each term as follows:
[tex]\[
35s^6 = 7s^2 \cdot 5s^4
\][/tex]
[tex]\[
7s^2 = 7s^2 \cdot 1
\][/tex]
3. Rewrite the polynomial: Using the common factor [tex]\( 7s^2 \)[/tex], we can rewrite the polynomial:
[tex]\[
35s^6 + 7s^2 = 7s^2 (5s^4 + 1)
\][/tex]
Therefore, the completely factored form of the polynomial [tex]\( 35s^6 + 7s^2 \)[/tex] is:
[tex]\[
7s^2 (5s^4 + 1)
\][/tex]
