Question 61·Easy·Equivalent Expressions

The coefficients have a greatest common factor of $5$. For the variable part, think about the smallest exponent of $y$ that appears in both terms.

Once you decide on the greatest common factor, divide each term by it to find what goes inside the parentheses. Then compare that factored form to the answer choices.

In Desmos, use $x$ instead of $y$. In the first line, type f(x) = 5x^4 - 15x^2 to represent the original expression.

On new lines, enter each choice with $x$ in place of $y$, for example: A(x) = 5x^2(x^2 - 3), B(x) = x^2(5x - 15), C(x) = 5x^2(x - 3), and D(x) = 15x^2(x^2 - 1).

Turn on all graphs. The correct choice will have a graph that lies exactly on top of the graph of $f(x)$ for all $x$-values. You can also create a table for $f(x)$ and for each choice (by clicking the gear icon and selecting "Table") and compare $y$-values at several $x$-values; the equivalent expression will match $f(x)$ at every tested point.

Look at the two terms in the expression: $5y^4$ and $-15y^2$.

• For the numbers $5$ and $15$, the greatest common factor is $5$.
• For the variables $y^4$ and $y^2$, the greatest common factor is the smaller power, $y^2$.

So the overall GCF of the expression is $5y^2$.

Now divide each term of $5y^4 - 15y^2$ by the GCF $5y^2$ to see what will go inside the parentheses.

• $5y^4 \div 5y^2 = y^2$
• $-15y^2 \div 5y^2 = -3$

So after factoring out $5y^2$, the factor that remains in parentheses is $y^2 - 3$.

Putting the GCF and the remaining factor together, you can write the original expression as

Now compare this factored expression with the answer choices and see that it exactly matches choice A.