To factor the polynomial [tex]\( 9w^6 + 12w^5 - 12w^4 \)[/tex] completely, we follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, we find the greatest common factor of all the terms in the polynomial. The terms are:
[tex]\[
9w^6, \ 12w^5, \ \text{and} \ -12w^4
\][/tex]
The GCF of the coefficients [tex]\( 9, 12, \text{and} -12 \)[/tex] is 3.
The GCF of [tex]\( w^6, w^5, \text{and} w^4 \)[/tex] is [tex]\( w^4 \)[/tex], since that is the highest power of [tex]\( w \)[/tex] that divides each term.
Therefore, the GCF of the entire polynomial is [tex]\( 3w^4 \)[/tex].
2. Factor out the GCF:
We factor [tex]\( 3w^4 \)[/tex] out of each term in the polynomial:
[tex]\[
9w^6 + 12w^5 - 12w^4 = 3w^4(3w^2 + 4w - 4)
\][/tex]
3. Factor the quadratic expression:
Next, we factor the quadratic expression [tex]\( 3w^2 + 4w - 4 \)[/tex]. To do this, we look for two numbers that multiply to [tex]\( 3 \times (-4) = -12 \)[/tex] and add to 4.
We find that [tex]\( 6 \times (-2) = -12 \)[/tex] and [tex]\( 6 + (-2) = 4 \)[/tex].
So, we rewrite the quadratic expression using these numbers:
[tex]\[
3w^2 + 4w - 4 = 3w^2 + 6w - 2w - 4
\][/tex]
4. Group and factor by grouping:
We group the terms in pairs and factor out the GCF from each pair:
[tex]\[
(3w^2 + 6w) + (-2w - 4) = 3w(w + 2) - 2(w + 2)
\][/tex]
Now, we factor out the common binomial factor [tex]\((w + 2)\)[/tex]:
[tex]\[
3w(w + 2) - 2(w + 2) = (3w - 2)(w + 2)
\][/tex]
5. Combine all factored terms:
Finally, we combine the GCF [tex]\( 3w^4 \)[/tex] with the factored form of the quadratic expression:
[tex]\[
9w^6 + 12w^5 - 12w^4 = 3w^4(3w - 2)(w + 2)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\( 9w^6 + 12w^5 - 12w^4 \)[/tex] is:
[tex]\[
\boxed{3w^4(3w - 2)(w + 2)}
\][/tex]
