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Answer :
To solve the problem of finding an equivalent expression for [tex]\(25t^2 - 15t + 60tv\)[/tex], we want to factor the given expression.
Let's break down the steps:
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients of the terms: [tex]\(25\)[/tex], [tex]\(-15\)[/tex], and [tex]\(60\)[/tex].
- The greatest common factor of these numbers is [tex]\(5\)[/tex].
- Additionally, each term contains a variable [tex]\(t\)[/tex], so we also factor out [tex]\(t\)[/tex].
2. Factor out the GCF:
- We factor [tex]\(5t\)[/tex] out of each term in the expression:
- From [tex]\(25t^2\)[/tex], factoring out [tex]\(5t\)[/tex] leaves us with [tex]\(5t\)[/tex].
- From [tex]\(-15t\)[/tex], factoring out [tex]\(5t\)[/tex] leaves us with [tex]\(-3\)[/tex].
- From [tex]\(60tv\)[/tex], factoring out [tex]\(5t\)[/tex] leaves us with [tex]\(12v\)[/tex].
3. Write the factored expression:
- After factoring out [tex]\(5t\)[/tex], the expression becomes:
[tex]\[
5t(5t - 3 + 12v)
\][/tex]
Reviewing the options provided:
- (C) [tex]\(5t(5t - 3 + 12v)\)[/tex] matches our factored expression.
Therefore, the correct choice is option (C), [tex]\(5t(5t - 3 + 12v)\)[/tex].
Let's break down the steps:
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients of the terms: [tex]\(25\)[/tex], [tex]\(-15\)[/tex], and [tex]\(60\)[/tex].
- The greatest common factor of these numbers is [tex]\(5\)[/tex].
- Additionally, each term contains a variable [tex]\(t\)[/tex], so we also factor out [tex]\(t\)[/tex].
2. Factor out the GCF:
- We factor [tex]\(5t\)[/tex] out of each term in the expression:
- From [tex]\(25t^2\)[/tex], factoring out [tex]\(5t\)[/tex] leaves us with [tex]\(5t\)[/tex].
- From [tex]\(-15t\)[/tex], factoring out [tex]\(5t\)[/tex] leaves us with [tex]\(-3\)[/tex].
- From [tex]\(60tv\)[/tex], factoring out [tex]\(5t\)[/tex] leaves us with [tex]\(12v\)[/tex].
3. Write the factored expression:
- After factoring out [tex]\(5t\)[/tex], the expression becomes:
[tex]\[
5t(5t - 3 + 12v)
\][/tex]
Reviewing the options provided:
- (C) [tex]\(5t(5t - 3 + 12v)\)[/tex] matches our factored expression.
Therefore, the correct choice is option (C), [tex]\(5t(5t - 3 + 12v)\)[/tex].
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