Question

Part A: The area of a square is (25x2 − 10x + 1) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work. (5 points) Part B: The area of a rectangle is (81x2 − 9y2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

Answer 1

Before getting started, let's look at notation. My guess is that by (25x2-10x+1) you mean "25 x squared plus 10x plus 1." Usually to show the exponent (x squared) we use  the ^ symbol so that means for Part A you are being asked to factor 25x^2 -10x + 1. We can also use a program like equation editor (part of Word) so that the expressions look like they would if you wrote then by hand in your notebook. I will do PART A using "^" and PART B using an editor that will make the notation look as it does in your notebook so you see both ways.

PART A:
As the expression given is the area of a square (and in a square the sides are all of equal length) when we factor (write as a product) this expression we should get two identical factors. If you remember how to distribute twice (teachers usually use the term F.O.I.L.) this question is asking you to go backwards...the expression is what you get when you "FOIL" so what were the two terms before?

25x^2 --> What can you multiply to get 25x^2? Since the terms have to be identical that's 5x. That is (5x)(5x) gives 25x^2.

Next we look at the last term (1) and think what can you multiply by itself to get 1? That's 1 as (1)(1)=1.

This means that the expression 25x^2 -10x + 1 = (5x+1)(5x+1) = (5x+1)^2.
The side of the square is 5x+1

PART B:
Here you are asked to factor $81 x^{2} -9 y^{2}$. You might notice that $81 x^{2}$ is a perfect square. $81 x^{2} =(9x)(9x)$. Similarly, $9 y^{2} =(3y)(3y)$. We are dealing with the difference (notice the subtraction sign between the terms) of squares.

How do we factor the difference of squares? It is: (square root of first term + square root of second term)(square root of first term - square term of second). In symbols it is: $a^{2} - b^{2} =(a+b)(a-b)$.

In our case, we get $81 x^{2} -9 y^{2} =(9x+3y)(9x-3y)$.

If you forget or don't see that it is the difference of squares you can still figure it out by reasoning it out. What two terms give you the first when multiplied and what two terms give you the second?