Let's break down the steps to solve the problem in detail:
### Finding the Greatest Common Factor (GCF)
Step 1: Calculate the GCF of [tex]\(14x^2 - 7x\)[/tex]
To find the GCF of the terms in [tex]\(14x^2 - 7x\)[/tex]:
- Factor each term:
- [tex]\(14x^2 = 2 \cdot 7 \cdot x \cdot x\)[/tex]
- [tex]\(-7x = -1 \cdot 7 \cdot x\)[/tex]
- Identify the common factors:
- The common factors in both terms are: [tex]\(7x\)[/tex]
Therefore, the GCF of [tex]\((14x^2 - 7x)\)[/tex] is [tex]\(7x\)[/tex].
Step 2: Calculate the GCF of [tex]\(6x - 3\)[/tex]
To find the GCF of the terms in [tex]\(6x - 3\)[/tex]:
- Factor each term:
- [tex]\(6x = 2 \cdot 3 \cdot x\)[/tex]
- [tex]\(-3 = -1 \cdot 3\)[/tex]
- Identify the common factors:
- The common factor in both terms is: [tex]\(3\)[/tex]
Therefore, the GCF of [tex]\((6x - 3)\)[/tex] is [tex]\(3\)[/tex].
### Finding the Common Binomial Factor
Step 3: Factor out the GCF from each expression
- From [tex]\(14x^2 - 7x\)[/tex]:
- Factor out [tex]\(7x\)[/tex]: [tex]\(14x^2 - 7x = 7x(2x - 1)\)[/tex]
- From [tex]\(6x - 3\)[/tex]:
- Factor out [tex]\(3\)[/tex]: [tex]\(6x - 3 = 3(2x - 1)\)[/tex]
We notice that the binomial [tex]\((2x - 1)\)[/tex] is common in both factorizations.
Thus, the common binomial factor is [tex]\(2x - 1\)[/tex].
### Factoring the Entire Expression
Step 4: Factor the entire expression [tex]\(14x^2 - 7x + 6x - 3\)[/tex]
- Combine like terms and identify common factors:
- The expression can be grouped into [tex]\((14x^2 - 7x) + (6x-3)\)[/tex]
- Factor out the common binomial factor:
- As we found earlier, [tex]\((14x^2 - 7x) = 7x(2x - 1) \)[/tex] and [tex]\((6x - 3) = 3(2x - 1)\)[/tex]
- Factor by grouping:
- [tex]\( 14x^2 - 7x + 6x - 3 = (7x + 3)(2x - 1)\)[/tex]
Therefore, the factored expression is [tex]\((2x - 1)(7x + 3)\)[/tex].
### Final Answers
- The GCF of the group [tex]\((14x^2 - 7x)\)[/tex] is [tex]\(7x\)[/tex].
- The GCF of the group [tex]\((6x - 3)\)[/tex] is [tex]\(3\)[/tex].
- The common binomial factor is [tex]\(2x - 1\)[/tex].
- The factored expression is [tex]\((2x - 1)(7x + 3)\)[/tex].
