Question
Given the expression: 4x2 + 18x − 10 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)
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Answer to a math question Given the expression: 4x2 + 18x − 10 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)
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**Part A:**
The first step in finding the greatest common factor (GCF) of the expression is to factor out any common factors from all the terms. In this case, there is no common numerical factor among the terms, but we can factor out the greatest common factor of the coefficients of x², x, and the constant term.
The coefficients of x², x, and the constant term are 4, 18, and -10 respectively. The factors of 4 are 1, 2, and 4. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 10 are 1, 2, 5, and 10. The GCF should be the highest common factor among these numbers, which is 2.
Therefore, the greatest common factor of the expression 4x² + 18x - 10 is 2.
**Part B:**
To factor the expression completely, we will first look for the factors of the quadratic term (4x²) and the constant term (-10) that sum up to the coefficient of the linear term (18x).
The expression is 4x² + 18x - 10.
The factors of 4x² are 4x and x. The factors of -10 are -2 and 5.
Checking the combination of factors, we have:
(4x - 2)(x + 5).
So, the factored form of 4x² + 18x - 10 is (4x - 2)(x + 5).
**Part C:**
To check our factoring, we multiply the two factors back together:
(4x - 2)(x + 5) = 4x*x + 4x*5 - 2*x - 2*5 = 4x² + 20x - 2x - 10 = 4x² + 18x - 10.
Therefore, our factoring is correct.
**Answer:**
- **Part A:** The greatest common factor is 2.
- **Part B:** The factored form is (4x - 2)(x + 5).
- **Part C:** The factoring is correct as (4x - 2)(x + 5) when multiplied back gives 4x² + 18x - 10.
The first step in finding the greatest common factor (GCF) of the expression is to factor out any common factors from all the terms. In this case, there is no common numerical factor among the terms, but we can factor out the greatest common factor of the coefficients of x², x, and the constant term.
The coefficients of x², x, and the constant term are 4, 18, and -10 respectively. The factors of 4 are 1, 2, and 4. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 10 are 1, 2, 5, and 10. The GCF should be the highest common factor among these numbers, which is 2.
Therefore, the greatest common factor of the expression 4x² + 18x - 10 is 2.
**Part B:**
To factor the expression completely, we will first look for the factors of the quadratic term (4x²) and the constant term (-10) that sum up to the coefficient of the linear term (18x).
The expression is 4x² + 18x - 10.
The factors of 4x² are 4x and x. The factors of -10 are -2 and 5.
Checking the combination of factors, we have:
(4x - 2)(x + 5).
So, the factored form of 4x² + 18x - 10 is (4x - 2)(x + 5).
**Part C:**
To check our factoring, we multiply the two factors back together:
(4x - 2)(x + 5) = 4x*x + 4x*5 - 2*x - 2*5 = 4x² + 20x - 2x - 10 = 4x² + 18x - 10.
Therefore, our factoring is correct.
**Answer:**
- **Part A:** The greatest common factor is 2.
- **Part B:** The factored form is (4x - 2)(x + 5).
- **Part C:** The factoring is correct as (4x - 2)(x + 5) when multiplied back gives 4x² + 18x - 10.
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