Once you know the greatest common factor, write the original expression as (GCF)$\times$(something). Then find what must go in the parentheses by dividing each term by the GCF.

If you are unsure, take a choice and distribute its outside factor through the parentheses to see if you get back $4t^{6}-20t^{4}+16t^{3}$. Only one choice will expand to the exact same expression.

In one line, type f(t) = 4t^6 - 20t^4 + 16t^3 to define the original function.

On new lines, type the expressions from the choices, for example g(t) = 4t^2(t^4 - 5t^2 + 2t), h(t) = 2t^3(2t^3 - 8t + 8), etc., each with a different letter.

Look at the graphs of $f(t)$ and each of the choice functions. The function whose graph lies exactly on top of (is indistinguishable from) the graph of $f(t)$ for all visible $t$ is the expression that is algebraically equivalent to the original.

Alternatively, create new lines with expressions like f(t) - g(t), f(t) - h(t), etc. The expression whose difference with $f(t)$ simplifies to a horizontal line at $y=0$ for all $t$ is the equivalent one.

Look at the coefficients and powers of $t$ in $4t^{6}-20t^{4}+16t^{3}$.

• Numerically, $\text{gcf}(4,20,16)=4$.
• For the variable part, each term has at least $t^{3}$ (since the exponents are $6,4,3$), so the common factor in $t$ is $t^{3}$.

So the overall greatest common factor is $4t^{3}$, and we can write the expression as $4t^{3} \times$ (something).

Now divide each original term by $4t^{3}$ to see what remains inside the parentheses:

So the three "leftover" terms that go inside the parentheses are $t^{3}$, $-5t$, and $4$. We have now completely factored out the GCF; the final step is to put this together and compare to the answer choices.

Putting the GCF and the leftover terms together gives the factored form

Distributing $4t^{3}$ back in confirms it matches the original expression:

This factored expression is exactly answer choice D, so D is the correct answer.