Factoring Polynomials – Quadratic Equations
Objective
Remember Pascal’s Triangle?
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
So, how does this help us understand binomials better?
Well, because, without knowing it at first, Pascal actually handed us a neat map to find what certain binomials will become given a certain power, without multiplying it out.
This sounds confusing, but let’s do an example:
Let’s start off with just squaring (a + b)
So:
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 2 1
Let’s start off with just cubing (a + b)
So:
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 3 3 1
Let’s keep going: (a + b) to the fourth power
So:
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 4 6 4 1
Last one: (a + b) to the fifth power
So:
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 5 10 10 5 1
SO, THAT’S HOW THE BINOMIAL THEOREM WORKS
By realizing that binomials multiply in the same way that Pascal’s triangle can be added, we can figure out what the answer is without actually doing the multiplication.
So, that is the Binomial theorem and how it works.
QUICK REVIEW OVER DIVISIBILITY RULES
So before we start factoring, it’s a really good idea to go over some divisibility rules, since we’re going to need them to factor.
So, let’s get started.
Numbers divisible by 2
A number is divisible by 2 if and only if it is even.
To be specific, a number is divisible by 2 if and only if the ones digit in the number is either a 0, 2, 4, 6 or an 8.
So a few examples of numbers that are divisible by 2:
2
14
268
3802
40246
524680
Numbers divisible by 3
A number is divisible by 3 if the sum of all of its digits are also divisible by 3.
So, a good example of this is:
582
We know it is divisible by 3 because:
5 + 8 + 2 = 15, and 15 is divisible by 3.
But let’s check:
Another example: 974238516
9 + 7 + 4 + 2 + 3 + 8 + 5 + 1 + 6 = 45
And we know that 4 + 5 = 9, and 9 is divisible by 3.
So 974238516 is divisible by 3.
(324746172 is the answer when you divide it.)
Numbers divisible by 4
A number is divisible by 4 if and only if it the ones digit and tens digit combined are divisible by 4.
An example of this would be:
352 is divisible by 4 because we know that 52 is divisible by 4.
But let’s check:
Another example:
7196 is divisible by 4 as well, because 96 is divisible by 4.
But, again, let’s check:
Numbers divisible by 5
A number is divisible by 5 if and only if its ones digit is 5 or 0.
So a few examples of numbers that are divisible by 5 are:
5
15
205
3550
40505
506060
7070705
Numbers divisible by 6
A number is divisible by 6 if and only if it is even and is divisible by 3.
Seems pretty obvious, but all you need to do is check if it is even, then add the digits together to see if it’s also divisible by 3.
So a few examples of numbers that are divisible by 6 are:
282
We know it’s even, and also 2 + 8 + 2 = 12 which is divisible by 3.
So it’s divisible by 6.
Another example:
5814
Again, an even number, so now we add the digits to see if it works:
5 + 8 + 1 + 4 = 18, which is divisible by 3.
So this number is divisible by 6.
NUMBERS DIVISIBLE BY 7
So, to be honest, finding out whether or not a number is divisible by 7 is actually harder than just dividing by 7. But for those of you who may think it’s easier to use the method, here it is:
To check if a number is divisible by 7: Take the last digit of the number, double it then subtract the result from the rest of the number. If the resulting number is evenly divisible by 7, so is the original number.
So an example of this would be:
343
3*2 = 6
So:
34 – 6 = 28
And of course, 28 is divisible by 7, so, so is 343.
Another example:
2401
1*2 = 2 and
240 – 2 = 238
8*2 = 16
23 – 16 = 7
And 7 is divisible by 7 (obviously) so 2401 is divisible by 7.
NUMBERS DIVISIBLE BY 9
A number is divisible by 9 if the sum of all of its digits are also divisible by 9.
So, a good example of this is:
585
We know it is divisible by 9 because:
5 + 8 + 5 = 18, and 18 is divisible by 9.
But let’s check:
Another example: 974238516
9 + 7 + 4 + 2 + 3 + 8 + 5 + 1 + 6 = 45
And we know that 4 + 5 = 9, and 9 is divisible by 9.
So 974238516 is divisible by 9.
(108248724 is the answer when you divide it.)
Factoring a quadratic
Example 1
Let’s say we are given:
We can see that it’s just 1, so there’s nothing to add to our factored equation.
However, we can see that we have 12 as our number.
So we break 12 into its multiples, and see if we can add two of those multiples together to get 7.
Well, we can see that
12 = 6 * 2
12 = 3 * 4
We know 6 + 2 = 8
But 3 + 4 = 7!
So, now we have our pieces!
Factoring the quadratic
So now we just plug in what we have:
So:
(x + )(x + )
And we know that 12 = 3 * 4, and 3 + 4 = 7, so:
(x + 3)(x + 4)
Now we solve!
(x + 3)(x + 4) = 0
x + 3 = 0
-3 -3
x = -3
And
x + 4 = 0
-4 -4
x = -4
So x = -3, -4!
So let’s try another one
Let’s say we’re given:
(x + )(x + )
And we know 2 = 2 * 1
And 2 + 1 = 3
So:
(x + 2)(x + 1) = 0
Now we solve!
x + 2 = 0
- 2 -2
x = -2
And
x + 1 = 0
-1 -1
x = -1
What about if there are negatives?
So the thing about factoring is, the quadratic that you are trying to factor actually gives you all of the information you need.
Whether you need to make a certain number negative or positive, as well as whether or not you need to subtract versus add, you’ll find it in the original quadratic.
You just have to know where to look:
WHAT A QUADRATIC TELLS US:
Let’s take a look at a sample quadratic:
Looking at this quadratic, we can see that the last number is negative:
So, this means that we are going to have alternating signs.
Or, in other words, we’re going to be subtracting.
The reason this works is because the only way to get a negative sign for the last number in our quadratic, is if the two numbers in parenthesis also have different signs.
We can also see that 3 is negative:
Which means the two numbers that are being subtracted, need to end up negative.
So, we’re going to find the factors of 4:
/\
4 1
2 2
And we’re going to subtract them to find what we need.
So:
- = 3
- = 0
As we can see, 4 – 1 = 3, so our factors are going to be:
(x – 4)(x + 1) = 0
Why is 4 negative?
Because we need to make a -3x
x – 4 = 0
x + 1 = 0
+ 4 + 4 -1 -1
x = 4 x = -1
WHAT A QUADRATIC ALSO TELLS US:
Let’s take a look at a sample quadratic:
Looking at this quadratic, we can see that the last number is positive:
So, this means that we are going to have the same sign.
Or, in other words, we’re going to be adding.
This is a little different than the last one.
Because the last number is a positive, this means we will be adding, and both numbers will need to be the same sign.
We can also see that 3 is negative:
Which means the two numbers that are being added, need to end up negative.
So, we’re going to find the factors of 2:
/\
2 1
And we’re going to add them to find what we need.
So:
+ = 3
As we can see, 2 + 1 = 3, so our factors are going to be:
(x – 2)(x - 1) = 0
Why are they both negative?
Because we need to make a -3x
x – 2 = 0
x - 1 = 0
+ 2 + 2 +1 +1
x = 2 x = 1
WHAT IF THERE’S SOMETHING NEXT TO X SQUARED?
Sometimes you’ll encounter something like:
What do we do?
Well, we factor both of them, and see if we can multiply those factors together to make the middle number.
It’s a little complicated, but looks something like this:
/ \
12 1
6 2
4 3
/ \
10 1
5 2
So now what we need to do is figure out which we can multiply together, then subtract, to get the number in the middle.
So:
12
10
12 * 10 = 120
1
1
1 * 1 = 1
-___
119
12 * 1 = 12
1 * 10 = 10
2
6
5
6 * 5 = 30
2
2
2 * 2 = 4
26
And there we have it!
So now, we need to make sure to make the 30 negative and 4 positive and we have out factors!
So:
(6x + 2)(2x – 5) = 0
Because:
-30x
+ 4x
= -26x
So now we set both equal to zero and solve for x:
6x + 2 = 0 2x – 5 = 0
-2 -2 +5 +5
6x = 2 2x = 5
_______ ________
6 6 2 2
SO IT CAN GET COMPLICATED
I mean, pretty complicated, but it’s usually the quickest way to solve quadratic equations.