To determine the factors of the polynomial [tex]\( x^3 + 11x^2 - 3x - 33 \)[/tex] by grouping, you can follow these steps:
1. Initial Grouping:
Start by splitting the polynomial into two groups:
[tex]\[
(x^3 + 11x^2) + (-3x - 33)
\][/tex]
2. Factor out the Greatest Common Factor (GCF) from each group:
* From the first group [tex]\( x^3 + 11x^2 \)[/tex], the GCF is [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 11)
\][/tex]
* From the second group [tex]\( -3x - 33 \)[/tex], the GCF is [tex]\( -3 \)[/tex]:
[tex]\[
-3(x + 11)
\][/tex]
3. Factor by Grouping:
Notice that [tex]\( (x + 11) \)[/tex] is a common factor in both terms:
[tex]\[
x^2(x + 11) - 3(x + 11)
\][/tex]
Thus, one way to determine the factors of [tex]\( x^3 + 11x^2 - 3x - 33 \)[/tex] by grouping is:
[tex]\[
x^2(x + 11) - 3(x + 11)
\][/tex]
Now let's check the given options:
1. [tex]\( x^2(x + 11) + 3(x - 11) \)[/tex]
2. [tex]\( x^2(x - 11) - 3(x - 11) \)[/tex]
3. [tex]\( x^2(x + 11) + 3(x + 11) \)[/tex]
4. [tex]\( x^2(x + 11) - 3(x + 11) \)[/tex]
Comparing these options with our factorization, the correct option that matches is:
[tex]\[
\boxed{x^2(x + 11) - 3(x + 11)}
\][/tex]
