Sure! Let's break down the problem step by step.
### Part A: Finding the Greatest Common Factor (3 points)
To find the greatest common factor (GCF) of the terms in the expression [tex]\(10x^2 + 25x - 15\)[/tex], we need to look at the coefficients: 10, 25, and -15.
1. List the factors of each coefficient:
- Factors of 10: [tex]\(1, 2, 5, 10\)[/tex]
- Factors of 25: [tex]\(1, 5, 25\)[/tex]
- Factors of -15: [tex]\(1, 3, 5, 15\)[/tex] (Only positive factors suffice for GCF calculation)
2. Identify the common factors:
- The common factors among 10, 25, and -15 are 1 and 5.
3. Greatest common factor:
- The greatest common factor is 5.
Answer for Part A: The greatest common factor is 5.
### Part B: Factoring the Expression Completely (5 points)
Given [tex]\(10x^2 + 25x - 15\)[/tex], we proceed as follows:
1. Factor out the GCF (which is 5):
[tex]\[
10x^2 + 25x - 15 = 5(2x^2 + 5x - 3)
\][/tex]
2. Factor the quadratic expression [tex]\(2x^2 + 5x - 3\)[/tex] inside the parentheses. We look for two numbers that multiply to [tex]\((2 \times -3 = -6)\)[/tex] and add up to 5.
- Possible factor pairs of -6: [tex]\((1, -6), (-1, 6), (2, -3), (-2, 3)\)[/tex]
- The pair that adds up to 5 is [tex]\(6\)[/tex] and [tex]\(-1\)[/tex].
3. Rewrite the middle term (5x) using 6 and -1:
[tex]\[
2x^2 + 5x - 3 = 2x^2 + 6x - x - 3
\][/tex]
4. Group the terms to factor by grouping:
[tex]\[
= (2x^2 + 6x) + (-x - 3)
\][/tex]
5. Factor each group:
[tex]\[
= 2x(x + 3) - 1(x + 3)
\][/tex]
6. Factor out the common binomial factor [tex]\(x + 3\)[/tex]:
[tex]\[
= (2x - 1)(x + 3)
\][/tex]
7. Combine the GCF factored out initially:
[tex]\[
10x^2 + 25x - 15 = 5(2x - 1)(x + 3)
\][/tex]
Answer for Part B: The completely factored form of [tex]\(10x^2 + 25x - 15\)[/tex] is [tex]\(5(2x - 1)(x + 3)\)[/tex].
### Part C: Checking Your Factoring by Multiplying (2 points)
To verify our factoring, we will multiply the factors back together and ensure we get the original expression:
1. Expand the binomials [tex]\((2x - 1)(x + 3)\)[/tex]:
[tex]\[
(2x - 1)(x + 3) = 2x(x + 3) - 1(x + 3)
\][/tex]
[tex]\[
= 2x^2 + 6x - x - 3
\][/tex]
[tex]\[
= 2x^2 + 5x - 3
\][/tex]
2. Multiply by the GCF (5):
[tex]\[
5(2x^2 + 5x - 3) = 10x^2 + 25x - 15
\][/tex]
We have obtained the original expression [tex]\(10x^2 + 25x - 15\)[/tex], which confirms that our factoring is correct.
Answer for Part C: The original expression [tex]\(10x^2 + 25x - 15\)[/tex] is recovered by multiplying the factors [tex]\(5(2x - 1)(x + 3)\)[/tex]. This verifies that our factoring is correct.
