To factor the polynomial [tex]\( -7x^3 + 21x^2 + 3x - 9 \)[/tex] by grouping, we need to follow a sequence of logical steps. Here's a detailed breakdown:
1. Rewrite the polynomial:
The given polynomial is [tex]\( -7x^3 + 21x^2 + 3x - 9 \)[/tex].
2. Group the terms:
Group the terms in pairs to facilitate factoring:
[tex]\[ (-7x^3 + 21x^2) + (3x - 9) \][/tex]
3. Factor out the common factor from each pair:
Identify the greatest common factor (GCF) in each group.
For the first pair [tex]\((-7x^3 + 21x^2)\)[/tex], the GCF is [tex]\(-7x^2\)[/tex]:
[tex]\[ -7x^2(x - 3) \][/tex]
For the second pair [tex]\((3x - 9)\)[/tex], the GCF is [tex]\(3\)[/tex]:
[tex]\[ 3(x - 3) \][/tex]
4. Factor by grouping:
Notice now that both groups contain the common factor [tex]\((x - 3)\)[/tex]:
[tex]\[ -7x^2(x - 3) + 3(x - 3) \][/tex]
Factor out the common binomial factor [tex]\((x - 3)\)[/tex]:
[tex]\[ (x - 3)(-7x^2 + 3) \][/tex]
5. Simplify the signs:
Rearrange the product to match standard formatting:
[tex]\[ -(x - 3)(7x^2 - 3) \][/tex]
Therefore, the factored form of the polynomial [tex]\( -7x^3 + 21x^2 + 3x - 9 \)[/tex] is:
[tex]\[
-(x - 3)(7x^2 - 3)
\][/tex]
So, the correct answer is:
[tex]\[
-(x - 3)(7x^2 - 3)
\][/tex]
