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The GCF of the group [tex](14x^2 - 7x)[/tex] is [tex]\square[/tex].

The GCF of the group [tex](6x - 3)[/tex] is [tex]\square[/tex].

The common binomial factor is [tex]\square[/tex].

The factored expression is [tex]\square[/tex].

Sagot :

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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].
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