SAT Equivalent Expressions Question 166 | Free SAT Prep | Aniko

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Question 166·Easy·Equivalent Expressions

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Which expression is equivalent to $5t(t - 2) - 3(t - 2)$ ?

For equivalent-expression questions involving algebraic expressions, first look for a greatest common factor (like a shared binomial) that you can factor out using the distributive property in reverse. If factoring directly is tricky, rewrite the expression in standard polynomial form, factor the polynomial, and then match it to the choices; as a quick check, you can also mentally or on scratch work expand one or two promising answer choices to see if they reproduce the original expression.

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Look for a common factor

Notice that both terms $5t(t - 2)$ and $-3(t - 2)$ contain the same binomial $(t - 2)$ . Think about how you can use that to factor the expression.

Recall the distributive property in reverse

The pattern $ax + bx = (a + b)x$ works in reverse too: if two terms share a factor, you can factor it out. How does this apply to $5t(t - 2) - 3(t - 2)$ ?

Compare factored forms to the choices

Once you factor the expression, check which option has the same factors (in any order). If expanding helps, you can quickly expand a choice to see if it matches the original expression.

Desmos Guide

Enter the original expression

In a Desmos expression line, type f(t) = 5*t*(t - 2) - 3*(t - 2) so you have a function representing the original expression.

Enter each answer choice as a separate expression

On new lines, enter:

A(t) = (t - 2)*(5*t - 3)
B(t) = (t - 2)*(5*t + 3)
C(t) = (t + 2)*(5*t - 3)
D(t) = 2*(t - 2)*(5*t - 3)

Compare values to see which matches

Create a table for t values (for example, $t = -2, -1, 0, 1, 2, 3$ ) and add columns for f(t), A(t), B(t), C(t), and D(t). The choice whose values always match f(t) for all tested $t$ is the expression equivalent to the original.

Step-by-step Explanation

Simplify the original expression

Start by distributing in each term to see the simplified polynomial:

5t(t - 2) - 3(t - 2) = 5t^2 - 10t - 3t + 6 = 5t^2 - 13t + 6

So the original expression is equivalent to $5t^2 - 13t + 6$ .

Factor the quadratic expression

Now factor $5t^2 - 13t + 6$ .

You need two numbers that:

multiply to $5 \cdot 6 = 30$ , and
add to $-13$ .

Those numbers are $-10$ and $-3$ , so split the middle term:

5t^2 - 13t + 6 = 5t^2 - 10t - 3t + 6

Next, group the terms to prepare for factoring by grouping:

5t^2 - 10t - 3t + 6 = (5t^2 - 10t) + (-3t + 6)

Factor by grouping and match to a choice

Factor each group:

(5t^2 - 10t) + (-3t + 6) = 5t(t - 2) - 3(t - 2)

Now factor out the common binomial $(t - 2)$ :

5t(t - 2) - 3(t - 2) = (t - 2)(5t - 3)

So the original expression is equivalent to $(t - 2)(5t - 3)$ , which corresponds to choice A.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation