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 solve the problem of finding which expression is equivalent to [tex]\(25t^2 - 15t + 60tv\)[/tex], we need to factor the expression by finding the greatest common factor.

1. Identify the terms:
The expression consists of three terms: [tex]\(25t^2\)[/tex], [tex]\(-15t\)[/tex], and [tex]\(60tv\)[/tex].

2. Find the greatest common factor (GCF):
- Look for the GCF of the coefficients: 25, 15, and 60. The greatest common factor of these numbers is 5.
- Next, look for common variables in all the terms. Here, each term contains at least one [tex]\(t\)[/tex].

Therefore, the GCF for the entire expression is [tex]\(5t\)[/tex].

3. Factor out the GCF:
- Divide each term by [tex]\(5t\)[/tex] and factor it out:
- [tex]\(25t^2 ÷ 5t = 5t\)[/tex]
- [tex]\(-15t ÷ 5t = -3\)[/tex]
- [tex]\(60tv ÷ 5t = 12v\)[/tex]

4. Rewrite the expression using the GCF:
So, the expression can be rewritten as:
[tex]\[
5t(5t - 3 + 12v)
\][/tex]

Therefore, the expression [tex]\(5t(5t - 3 + 12v)\)[/tex] is equivalent to the given expression [tex]\(25t^2 - 15t + 60tv\)[/tex].

The correct answer is:
(C) [tex]\(5t(5t - 3 + 12v)\)[/tex]

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