High School

Which expression is equivalent to [tex]$25 t^2 - 15 t + 60 t v$[/tex]?

A. [tex]$25(t^2 - 15 t + 60 t v)$[/tex]
B. [tex]$5 t(5 t^2 - 3 t + 12 t v)$[/tex]
C. [tex]$5 t(5 t - 3 + 12 v)$[/tex]
D. [tex]$25 t(t - 15 + 60 v)$[/tex]

Answer :

To determine which expression is equivalent to [tex]\(25t^2 - 15t + 60tv\)[/tex], we need to factor out the greatest common factor from the expression.

1. Identify the common factors in the terms:
- The terms are [tex]\(25t^2\)[/tex], [tex]\(-15t\)[/tex], and [tex]\(60tv\)[/tex].
- Each term has at least a [tex]\(5t\)[/tex] as a factor.

2. Factor out the greatest common factor, [tex]\(5t\)[/tex]:
- When we factor [tex]\(5t\)[/tex] out of each term, we divide each term by [tex]\(5t\)[/tex]:
[tex]\[
25t^2 \div 5t = 5t
\][/tex]
[tex]\[
-15t \div 5t = -3
\][/tex]
[tex]\[
60tv \div 5t = 12v
\][/tex]

3. Write the factored expression:
- After factoring [tex]\(5t\)[/tex] from each term, the expression becomes:
[tex]\[
5t(5t - 3 + 12v)
\][/tex]

4. Match the factored expression to the choices provided:
- Let's look at the choices:
- (A) [tex]\(25(t^2 - 15t + 60tv)\)[/tex]
- (B) [tex]\(5t(5t^2 - 3t + 12tv)\)[/tex]
- (C) [tex]\(5t(5t - 3 + 12v)\)[/tex]
- (D) [tex]\(25t(t - 15 + 60v)\)[/tex]

- From the factorization we did, we see that choice (C) [tex]\(5t(5t - 3 + 12v)\)[/tex] is equivalent to the original expression.

Therefore, the correct option is (C) [tex]\(5t(5t - 3 + 12v)\)[/tex].

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