To factor the expression [tex]\(2m^3 - 26m^2 + 80m\)[/tex], follow these steps:
1. Factor out the Greatest Common Factor (GCF):
The given expression is [tex]\(2m^3 - 26m^2 + 80m\)[/tex]. First, identify the greatest common factor in all terms, which is [tex]\(2m\)[/tex]. Factoring out [tex]\(2m\)[/tex], we get:
[tex]\[
2m(m^2 - 13m + 40)
\][/tex]
2. Factor the quadratic expression:
Now, we need to factor the quadratic expression [tex]\(m^2 - 13m + 40\)[/tex]. We are looking for two numbers that multiply to [tex]\(40\)[/tex] (the constant term) and add up to [tex]\(-13\)[/tex] (the coefficient of the linear term).
The two numbers that satisfy these conditions are [tex]\(-8\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[
-8 \times -5 = 40
\][/tex]
[tex]\[
-8 + -5 = -13
\][/tex]
3. Express the quadratic as a product:
We can now write the quadratic expression as a product of two binomials:
[tex]\[
m^2 - 13m + 40 = (m - 8)(m - 5)
\][/tex]
4. Combine all factors:
Substitute this back into the factored form:
[tex]\[
2m(m - 8)(m - 5)
\][/tex]
So, the fully factored form of the expression [tex]\(2m^3 - 26m^2 + 80m\)[/tex] is:
[tex]\[
2m(m - 8)(m - 5)
\][/tex]
From the given options, the correct one is:
[tex]\[
2m(m - 8)(m - 5)
\][/tex]
