Let's factor the polynomial [tex]\( 14x^2 + 6x - 7x - 3 \)[/tex] by grouping. We will go through the process step-by-step.
1. Group the polynomial terms:
[tex]\[ 14x^2 + 6x - 7x - 3 = (14x^2 - 7x) + (6x - 3) \][/tex]
2. Find the Greatest Common Factor (GCF) of each group:
- First group [tex]\( (14x^2 - 7x) \)[/tex]:
[tex]\( 14x^2 \)[/tex] and [tex]\( -7x \)[/tex] both have a common factor of [tex]\( 7x \)[/tex].
So, the GCF of [tex]\( (14x^2 - 7x) \)[/tex] is [tex]\( 7x \)[/tex].
- Second group [tex]\( (6x - 3) \)[/tex]:
[tex]\( 6x \)[/tex] and [tex]\( -3 \)[/tex] both have a common factor of [tex]\( 3 \)[/tex].
So, the GCF of [tex]\( (6x - 3) \)[/tex] is [tex]\( 3 \)[/tex].
3. Factor out the GCF from each group:
- First group [tex]\( (14x^2 - 7x) \)[/tex]:
Factor out [tex]\( 7x \)[/tex]:
[tex]\[ 14x^2 - 7x = 7x(2x - 1) \][/tex]
- Second group [tex]\( (6x - 3) \)[/tex]:
Factor out [tex]\( 3 \)[/tex]:
[tex]\[ 6x - 3 = 3(2x - 1) \][/tex]
4. Combine the groups to factor by grouping:
[tex]\[ 14x^2 + 6x - 7x - 3 = 7x(2x - 1) + 3(2x - 1) \][/tex]
5. Find the common binomial factor:
We notice that both terms have a common binomial factor [tex]\( (2x - 1) \)[/tex].
6. Factor out the common binomial factor:
[tex]\[ 7x(2x - 1) + 3(2x - 1) = (7x + 3)(2x - 1) \][/tex]
So, the completed statements are:
The GCF of the group [tex]\( (14x^2 - 7x) \)[/tex] is [tex]\( \boxed{7x} \)[/tex].
The GCF of the group [tex]\( (6x - 3) \)[/tex] is [tex]\( \boxed{3} \)[/tex].
The common binomial factor is [tex]\( \boxed{2x - 1} \)[/tex].
The factored expression is [tex]\( \boxed{(7x + 3)(2x - 1)} \)[/tex].
