Answer :
Sure! Let's factor out the negative sign with the greatest common factor (GCF) from the expression [tex]\(-4x^3 + 28x^2 - 20x\)[/tex].
### Step-by-Step Solution:
1. Identify the common factors:
Each term in the expression [tex]\(-4x^3 + 28x^2 - 20x\)[/tex] has a factor of [tex]\(x\)[/tex]. The coefficients are [tex]\(-4\)[/tex], [tex]\(28\)[/tex], and [tex]\(-20\)[/tex].
2. Find the greatest common factor of the coefficients:
The coefficients are [tex]\(-4\)[/tex], [tex]\(28\)[/tex], and [tex]\(-20\)[/tex]. Let's find their GCF:
- Factors of [tex]\(4\)[/tex]: [tex]\(1, 2, 4\)[/tex]
- Factors of [tex]\(28\)[/tex]: [tex]\(1, 2, 4, 7, 14, 28\)[/tex]
- Factors of [tex]\(20\)[/tex]: [tex]\(1, 2, 4, 5, 10, 20\)[/tex]
The GCF of [tex]\(-4\)[/tex], [tex]\(28\)[/tex], and [tex]\(-20\)[/tex] is [tex]\(4\)[/tex].
3. Factor out [tex]\(-4x\)[/tex]:
Since we are factoring out a negative term, we'll factor out [tex]\(-4x\)[/tex]:
[tex]\[
-4x^3 + 28x^2 - 20x = -4x(x^2 - 7x + 5)
\][/tex]
4. Check by distribution:
To ensure correctness, we can distribute [tex]\(-4x\)[/tex] back through the expression inside the parentheses:
[tex]\[
-4x(x^2) = -4x^3, \quad -4x(-7x) = 28x^2, \quad -4x(5) = -20x
\][/tex]
Everything checks out! Therefore, the expression in factored form is:
[tex]\[
-4x(x^2 - 7x + 5)
\][/tex]
### Step-by-Step Solution:
1. Identify the common factors:
Each term in the expression [tex]\(-4x^3 + 28x^2 - 20x\)[/tex] has a factor of [tex]\(x\)[/tex]. The coefficients are [tex]\(-4\)[/tex], [tex]\(28\)[/tex], and [tex]\(-20\)[/tex].
2. Find the greatest common factor of the coefficients:
The coefficients are [tex]\(-4\)[/tex], [tex]\(28\)[/tex], and [tex]\(-20\)[/tex]. Let's find their GCF:
- Factors of [tex]\(4\)[/tex]: [tex]\(1, 2, 4\)[/tex]
- Factors of [tex]\(28\)[/tex]: [tex]\(1, 2, 4, 7, 14, 28\)[/tex]
- Factors of [tex]\(20\)[/tex]: [tex]\(1, 2, 4, 5, 10, 20\)[/tex]
The GCF of [tex]\(-4\)[/tex], [tex]\(28\)[/tex], and [tex]\(-20\)[/tex] is [tex]\(4\)[/tex].
3. Factor out [tex]\(-4x\)[/tex]:
Since we are factoring out a negative term, we'll factor out [tex]\(-4x\)[/tex]:
[tex]\[
-4x^3 + 28x^2 - 20x = -4x(x^2 - 7x + 5)
\][/tex]
4. Check by distribution:
To ensure correctness, we can distribute [tex]\(-4x\)[/tex] back through the expression inside the parentheses:
[tex]\[
-4x(x^2) = -4x^3, \quad -4x(-7x) = 28x^2, \quad -4x(5) = -20x
\][/tex]
Everything checks out! Therefore, the expression in factored form is:
[tex]\[
-4x(x^2 - 7x + 5)
\][/tex]
