Given the expression: [tex]$10x^2 + 25x - 15$[/tex]

Part A: What is the greatest common factor? Explain how to find it. (3 points)

Part B: Factor the expression completely. Show all necessary steps. (5 points)

Part C: Check your factoring from Part B by multiplying. Show all necessary steps. (2 points)



Answer :

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.