To solve the equation [tex]\( n^3 - 9n^2 - n + 9 = 0 \)[/tex] using factoring by grouping, follow these steps:
1. Group the terms:
Separate the polynomial into two groups:
[tex]\[
(n^3 - 9n^2) + (-n + 9)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
From the first group [tex]\( n^3 - 9n^2 \)[/tex], the GCF is [tex]\( n^2 \)[/tex]. Factor it out:
[tex]\[
n^2(n - 9)
\][/tex]
From the second group [tex]\( -n + 9 \)[/tex], the GCF is [tex]\(-1\)[/tex]. Factor it out:
[tex]\[
-1(n - 9)
\][/tex]
Putting this together, we have:
[tex]\[
n^2(n - 9) - 1(n - 9) = 0
\][/tex]
3. Factor by grouping:
Notice that [tex]\( (n - 9) \)[/tex] is a common factor in both terms:
[tex]\[
(n^2 - 1)(n - 9) = 0
\][/tex]
4. Factor the quadratic expression:
The expression [tex]\( n^2 - 1 \)[/tex] is a difference of squares and can be factored further:
[tex]\[
(n + 1)(n - 1)(n - 9) = 0
\][/tex]
5. Solve for [tex]\( n \)[/tex]:
Set each factor to zero and solve for [tex]\( n \)[/tex]:
[tex]\[
n + 1 = 0 \quad \Rightarrow \quad n = -1
\][/tex]
[tex]\[
n - 1 = 0 \quad \Rightarrow \quad n = 1
\][/tex]
[tex]\[
n - 9 = 0 \quad \Rightarrow \quad n = 9
\][/tex]
Thus, the solutions to the equation [tex]\( n^3 - 9n^2 - n + 9 = 0 \)[/tex] are:
[tex]\[
\boxed{-1, 1, 9}
\][/tex]
