factor expressions

Question

How do I completely factor this expression?

 

t^2 + 4tv + 4v^2

Answer 1

Hello!

 

We can factor this the same way we would factor any other polynomial such as x² + 4x + 4. The added variable is only there to try and trick you. 

 

Here's how you would do it:

 

Because the highest power of each variable is to the second & all terms are positive, we know the answer will be in form ( ? + ? ) * ( ? + ? )

 

The first term in each part of the expansion has to be t because when we check our work, we will notice that F.O.I.L. tells us to multiply t to t to get t^2. (This is further explained in the check your work section).

 

So far, we know the answer is ( t + ? ) * ( t + ? ).

 

Now to find the final unknown numbers. Let's only focus on the co-efficient part for now: the 4 and the other 4. We know the last two numbers must equal 4 when multiplied because of F.O.I.L. and 4 when added because of F.O.I.L. These numbers can only be 2. 

 

Now that we know the unknown number, we can't plug it into the expansion form just yet. Remember when I said to only focus on the co-efficient? Well, now that we've found it, we have to remember to recognize the variable that belongs with it. Because of F.O.I.L. we need two numbers that equal 4v². Well, we found the co-efficient to be 2, now all we have to do is place a v to each one.

 

It will look like this: ( t + 2v ) * ( t + 2v )

 

Always remember to check your work!:

 

F.O.I.L.: t²

F.O.I.L.: t² + 2tv

F.O.I.L.: t² + 2tv + 2tv

F.O.I.L.: t² + 2tv + 2tv + 4v²

Simplify: t² + 4tv + 4v²

 

This proves that (t+2v)(t+2v) is the completely factored form of t² + 4tv + 4v².

 

I hope this helps and let me know if you have any further questions! Happy New Years! :)