Let's tackle the problem step-by-step:
Part A: Finding the Greatest Common Factor (GCF)
Given the expression: [tex]\(12x^2 + 32x - 12\)[/tex].
To find the GCF, we first consider the coefficients of the terms: 12, 32, and -12. The GCF is the largest number that can divide all of them.
- The factors of 12 are: 1, 2, 3, 4, 6, 12.
- The factors of 32 are: 1, 2, 4, 8, 16, 32.
- The factors of -12 are: 1, 2, 3, 4, 6, 12.
The common factors among these numbers are: 1, 2, 4.
Therefore, the greatest common factor of 12, 32, and -12 is 4.
[tex]\[
\text{GCF} = 4
\][/tex]
Part B: Factoring the Expression Completely
To factor the expression [tex]\(12x^2 + 32x - 12\)[/tex] completely:
1. Step 1: Factor out the GCF
We divide each term in the expression by the GCF (4):
[tex]\[
12x^2 \div 4 = 3x^2, \quad 32x \div 4 = 8x, \quad -12 \div 4 = -3
\][/tex]
So, we have:
[tex]\[
12x^2 + 32x - 12 = 4(3x^2 + 8x - 3)
\][/tex]
2. Step 2: Factor the quadratic expression [tex]\(3x^2 + 8x - 3\)[/tex]
We need to find two numbers that multiply to the product of [tex]\(a \cdot c\)[/tex] (the coefficient of [tex]\(x^2\)[/tex] and the constant term) and add to [tex]\(b\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
- Here, [tex]\(a = 3\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = -3\)[/tex].
- We need two numbers that multiply to [tex]\((3 \cdot -3 = -9)\)[/tex] and add up to 8.
These numbers are 9 and -1.
3. Step 3: Rewrite the middle term using these numbers
[tex]\[
3x^2 + 9x - x - 3
\][/tex]
4. Step 4: Factor by grouping
Group the terms to facilitate factoring:
[tex]\[
(3x^2 + 9x) + (-x - 3)
\][/tex]
Factor out the common factors in each group:
[tex]\[
3x(x + 3) - 1(x + 3)
\][/tex]
Now, we can factor out the common factor [tex]\((x + 3)\)[/tex]:
[tex]\[
(3x - 1)(x + 3)
\][/tex]
So, the completely factored expression is:
[tex]\[
4(3x - 1)(x + 3)
\][/tex]
Part C: Check the Factoring by Multiplying
To ensure our factored form is correct, we will multiply the factors back together and check if we get the original expression.
1. Multiply [tex]\((3x - 1)\)[/tex] and [tex]\((x + 3)\)[/tex]:
[tex]\[
(3x - 1)(x + 3) = 3x(x) + 3x(3) - 1(x) - 1(3)
\][/tex]
[tex]\[
= 3x^2 + 9x - x - 3
\][/tex]
[tex]\[
= 3x^2 + 8x - 3
\][/tex]
2. Multiply the result by the GCF (4):
[tex]\[
4 \cdot (3x^2 + 8x - 3) = 12x^2 + 32x - 12
\][/tex]
Thus, we have confirmed that the original expression [tex]\(12x^2 + 32x - 12\)[/tex] was factored correctly as [tex]\(4(3x - 1)(x + 3)\)[/tex].
