If there is a constant in the math universe, it is the letter X. In algebra, X is the source of many a mental math meltdown. “Finding ” is the instruction in every single problem. Learn here to factor polynomials.
But X can be fun! We can play with it, move it around, multiply it, divide it, and cancel it out. X really lets us see the relationship between numbers and how math theorems work.
Totally Awesome, I know!
But I know the only reason you’re here is to learn about polynomials and what to do with them.
Factoring polynomials is a crucial step to using the quadratic formula. Factoring polynomials is easier, though, and faster, too.
Let’s look at some X factors.
3x, x, -x2, 5x2, 5x, 7x, 20, -4
The first thing we do is put them in descending order of x.
-x2, 5x2, 3x, x, 5x, 7x, 20, -4
We can simplify them by adding like-factors together: all of the x2, then all the x, then the numbers without x
4x2 + 16x + 16
The next step is to look for common factors in all three groups. For example, these groups all have even numbered coefficients, so we know that 2 is a factor. Even more significant, we can see that all these integers are divisible by 4!
Take out the four from all of them.
4(x2 + 4x + 4)
We’re almost there!
Let’s look at (x2 + 4x + 4) . Can we break this down anymore?
The answer is yes! With the FOIL system!
FOIL stands for “first, outside, inside, last”. This is a cute way of saying that we got this expression by multiplying two parts separate factors together. Each of these factors is called a Binomial. (This is because each factor has two parts: an unknown and a number).
Here we are:
(a + b) X (c + d) = (x2 + 4x + 4) .
Basically, we end up with: ac+ ad + bc + bd. When we add them all together, we get (x2 + 4x + 4).
In this case, bd = 4, the last part of our expression, while ac = x2, the first part of our expression. The other two parts, ad+ bc, need to equal 4x.
We know that ac = x2, so A and C must be x. So that’s easy.
The hard part is figuring out ad and bc. But to solve that, we can look to bd. What are factors that, when multiplied together, give us 4? 4 breaks down to 1, 2, and 4. That means B or D must be 1, 2, or 4.
*****Let’s try 1 and 4. Plug them in and see what happens
(x + 1)(x + 4)
Use FOIL to multiply them together: First, Outside, Inside, Last.
We get x2 + 4x + x + 4
Add them together: x2 + 5x + 4 ≠ x2 + 4x + 4
Yikes! That didn’t work.****
Then it must be 2. This actually makes sense, since 2 + 2 = 4 or 2x + 2x = 4x!
Let’s double check
(x +2)(x + 2) → x2 + 2x + 2x + 4 → x2 + 4x + 4 Bravo!
We can even make this simpler. Since (x + 2) is the same, we can just square the binomial (x + 2)2
So the final answer is 4(x + 2)2
Here’s a rough one: 3x3 + 15x2 – 36x
First thing is to take out the greatest common factor: 3x Check it and see!
3x(x2 + 5x – 12)
Now, we can FOIL the larger expression.
****Here’s a trick: look at the coefficients in the middle (5) and end (12). You know that two factors multiplied together will bring you the end number, but they must be added/subtracted to form the middle number 5.
What are the multiplication factors of 12? 1 and 12, 2 and 6, and 3 and 4. None of these sets of factors, when added or subtracted, give you 5.
If you don’t believe me, try finding the binomials through trial and error. You’ll be at it forever!
Rats! We’ve been “foiled!” We can’t break this down further!
The answer is
3x(x2 + 5x – 12)
One more: -2x2 + 17x – 36
First of all, are there any common factors? Unfortunately, no. One number is odd, so 2 is not a factor in all three. The last number has no x, so that can’t be a factor, either.
We’ll just proceed to the FOIL, then.
This is a bit hard, since we have THREE GROUPS of numbers to deal with: 2, 13, 36. But we can still do this! Just take your time.
What are the factors of 36 that can add up to 36?
36 is broken down to: 1 and 36, 2 and 18, 3 and 12, 4, and 9, and 6 and 6.
None of these really add up to 17, but we have to consider the 2. There are two factors that make up 2: 1 and 2. Can we multiply any of the factors that make up 36 by 1 and 2, and then add them together to get 17?
Look at factors 4 and 9. They match up with 1 and 2. Multiply and add the sums:
1 x 9 = 9
2 x 4 = 8
9 + 8 = 17!!! Hooray!
Since the 2 and the 4 are multiplied together, they must be in separate factors: (2x, 9)(x, 4)
Now to determine which ones are positive and negative.
This might sound harder than it is.
We know that two positives multiplied together form a positive. (5 x 4) = 20
Two negatives multiplied together form a positive. (-7 x -9) = 63
A positive and negative multiplied together form a negative. (-8 x 2) = -16
Go back to the original equation and our factors:
-2x2 + 17x – 36
(2x, 9)(x, 4)
We know the first and last coefficients are negative. That means one integer in each binomial must be negative and one must be positive so that, when multiplied, we get negative numbers. But we also know that the middle coefficient is positive.
To get the middle coefficient we need to make sure they are positive. That’s the only way they can ADD up to a positive 17. We need a 9x and an 8x. We can multiply two positives together or two negatives together to get positives.
Here’s the answer:
(-2x + 9)(x – 4) We can use trial and error by multiplying all the factors and adding ‘em up.
-2x2, 8x, 9x, -36.
-2x2 + 17x – 36
(-2x + 9)(x – 4)