Answer :
To solve the problem of finding which expression is equivalent to [tex]\(25 t^2 - 15 t + 60 t v\)[/tex], we need to factor the expression. Let's go through the steps:
1. Identify the Common Factor:
Look for the greatest common factor (GCF) in all of the terms: [tex]\(25 t^2, -15 t,\)[/tex] and [tex]\(60 t v\)[/tex].
2. Factor Out the GCF:
The terms all include [tex]\(t\)[/tex], and the numerical parts can be divided by 5. Therefore, the GCF is [tex]\(5t\)[/tex].
3. Divide Each Term by the GCF:
- Divide [tex]\(25 t^2\)[/tex] by [tex]\(5t\)[/tex]:
[tex]\[
\frac{25 t^2}{5 t} = 5 t
\][/tex]
- Divide [tex]\(-15 t\)[/tex] by [tex]\(5t\)[/tex]:
[tex]\[
\frac{-15 t}{5 t} = -3
\][/tex]
- Divide [tex]\(60 t v\)[/tex] by [tex]\(5t\)[/tex]:
[tex]\[
\frac{60 t v}{5 t} = 12 v
\][/tex]
4. Write the Factored Expression:
By grouping these results together, the expression can be rewritten as:
[tex]\[
5 t (5 t - 3 + 12 v)
\][/tex]
Thus, the expression that is equivalent to [tex]\(25 t^2 - 15 t + 60 t v\)[/tex] is option (C): [tex]\(5 t(5 t - 3 + 12 v)\)[/tex].
1. Identify the Common Factor:
Look for the greatest common factor (GCF) in all of the terms: [tex]\(25 t^2, -15 t,\)[/tex] and [tex]\(60 t v\)[/tex].
2. Factor Out the GCF:
The terms all include [tex]\(t\)[/tex], and the numerical parts can be divided by 5. Therefore, the GCF is [tex]\(5t\)[/tex].
3. Divide Each Term by the GCF:
- Divide [tex]\(25 t^2\)[/tex] by [tex]\(5t\)[/tex]:
[tex]\[
\frac{25 t^2}{5 t} = 5 t
\][/tex]
- Divide [tex]\(-15 t\)[/tex] by [tex]\(5t\)[/tex]:
[tex]\[
\frac{-15 t}{5 t} = -3
\][/tex]
- Divide [tex]\(60 t v\)[/tex] by [tex]\(5t\)[/tex]:
[tex]\[
\frac{60 t v}{5 t} = 12 v
\][/tex]
4. Write the Factored Expression:
By grouping these results together, the expression can be rewritten as:
[tex]\[
5 t (5 t - 3 + 12 v)
\][/tex]
Thus, the expression that is equivalent to [tex]\(25 t^2 - 15 t + 60 t v\)[/tex] is option (C): [tex]\(5 t(5 t - 3 + 12 v)\)[/tex].
