To factor the polynomial [tex]\( 36x^4 + 18x^3 + 40x^2 \)[/tex] completely, we should follow these steps:
1. Look for the greatest common factor (GCF) of all the terms. Here, each term has a factor of [tex]\( x^2 \)[/tex], so the GCF is [tex]\( x^2 \)[/tex].
2. Factor out [tex]\( x^2 \)[/tex] from each term in the polynomial:
[tex]\[
36x^4 + 18x^3 + 40x^2 = x^2(36x^2 + 18x + 40)
\][/tex]
3. Now, we look to factor the quadratic polynomial inside the parentheses, [tex]\( 36x^2 + 18x + 40 \)[/tex]:
After factoring out the GCF of the quadratic polynomial, the polynomial becomes [tex]\( 2(18x^2 + 9x + 20) \)[/tex].
4. So, we can combine these factors:
[tex]\[
36x^4 + 18x^3 + 40x^2 = 2x^2(18x^2 + 9x + 20)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\( 36x^4 + 18x^3 + 40x^2 \)[/tex] is:
[tex]\[
2x^2 (18x^2 + 9x + 20)
\][/tex]
