At a Glance - Recognizing Products
Sometimes we can tell by looking at a polynomial that it's a product of a particular type. Not that we want to put polynomials into a box, but...it does preserve their freshness.
Difference of Two Squares
Finding the difference of two squares is a two-way street. When we multiply a sum and difference of the same terms together, we have a difference of two squares. If you put your car in reverse and look in your rearview mirror, you see that a difference of two squares comes from the product of a sum and difference of the same terms. Now, no more algebra while driving. It isn't safe.
Any polynomial that looks like ▲2 – ■2 is the result of the polynomial multiplication (▲ + ■)(▲ – ■).
To factor the polynomial, we write it as the product of the sum and the difference.
Factor the polynomial x2 – 9.
This is a difference of two squares, so we know it can be factored as (▲ + ■)(▲ – ■) where ▲2 = x2 and ■2 = 9.
Therefore, we can take ▲ = x and ■ = 3 and factor the polynomial like so:
(x + 3)(x – 3)
See ya later, shapes. Go find a preschooler to play with.
Factor the polynomial 16y4 – x4.
The variables have exponents greater than 2, but this polynomial is still a difference of two squares. It's trying really hard not to be recognized. Probably because it doesn't want the paparazzi all up in its face.
Since 16y4 – x4 = (4y2)2 – (x2)2, we can factor the polynomial as:
(4y2 + x2)(4y2 – x2)
Beautiful. No cameras, please.
Sometimes you can figure out the factors of a polynomial by looking at it. If you have advanced telepathic powers, you may not even need to do that much.
We know what happens when we square a binomial, so we can recognize polynomials that are squares of binomials. Then we can work backwards to find the factors. If you want to challenge yourself, try working backwards, upside-down, and submerged underwater. We'll spot you.
Here's a quick reminder of what our perfect-square trinomials look like:
(▲ + ■)2 = ▲2 + 2▲■ + ■2
(▲ – ■)2 = ▲2 – 2▲■ + ■2
Factor the polynomial 4y2 + 12y + 9.
The second term is positive, so we'll see if we can get this polynomial by squaring a binomial of the form (▲ + ■)2.
We need ▲2 = 4y2, so ▲ = 2y is a good choice; nay, the choice of champions.
We also need ■2 = 9, so take ■ = 3.
It seems like the binomial we want is 2y + 3, but we need to check to make sure the middle term comes out right. The middle is important. If the ends were all that mattered, feature-length films would be a whole lot shorter.
When we square 2y + 3, we have a middle term of 2(2y)(3). Since 2(2y)(3) = 12y, the middle term of our original polynomial, our final answer is indeed:
4y2 + 12y + 9 = (2y + 3)2
Whew. Beginning? Check. Middle? Check. End? Check. We just got ourselves either a correctly factored polynomial or the rough draft of a perfectly structured screenplay. See if you can get James Cameron on the horn.
Is there a binomial whose square equals 25x2 + 10x + 4? If so, what is it?
Let's see if this polynomial is the square of a binomial (▲ + ■)2.
We would need ▲2 = 25x2, so ▲ = 5x.
We also need ■2 = 4, so take ■ = 2.
It seems like the binomial we want is 5x + 2.
When we square 5x + 2, however, the middle term will be 2(5x)(2) = 20x. While we're sure 20x is a perfectly nice term and we wish it no ill will, it isn't what we were looking for since it differs from the middle term in our original polynomial. Sadly, or perhaps not that sadly, we can't have any binomial whose square is 25x2 + 10x + 4. We would go back to the drawing board if we had one. Ooh...we should put that on our birthday list.
Factor the polynomial 3y4 – 27x2.
Factor the polynomial 8x2 – 56x + 98.
Factor 16x2 – 25.
Factor x2y2 – x2.
Factor 5y6 – 5x4.
Factor 9x4 + 24x2 + 16 if possible.
Factor 25x2 – 30x + 9 if possible.
Factor 3x2 – 12xy + 12y2 if possible.
Factor 4x6 + 24x3 + 36 if possible.
Factor x4 + 7x2 + 49 if possible.
Factor 9x2 – 8x + 16 if possible.