Blog 4 Jeff Greenberg

Jeff Greenberg

Blog 4

FACTORING

Factoring Algebraic Equations

a(b+c)= ab+ac

Factoring is an important exercise in mathematics. In its simplest form, lets say we have the algebraic equation y=12X+6. How would we factor this?

Well, we need to find a common factor of all terms in the equation y=12X+6.

We see that 6 is divisible by 2 and 12 is divisible by 2.

If we divide both terms by 2 and put a 2 outside the parenthesis, we get the equation y=2(6X+3). If we distribute the 2 to both terms (multiply 2 by both terms inside the parenthesis) we get y=12X+6.

So, we learned that both 6 and 12 are divisible by 2. We also learned that 6/2=3 and 12/2=6.

That being said, we pulled a 2 outside and got the factored equation y=2(6X+3).

Factoring Quadratic Equations

When factoring a quadratic equation, we are basically finding a way to make the expression (X^2+aX+bX+ab) into the expression (X+a)(X+b) and vice versa.

The form (X+a)(X+b)= X^2+aX+bX+ab

How did I factor the left side of this into a full equation on the right?

Well, we first need to learn to acronym “FOIL”.

FOIL

“FOIL” stands for: First, Outer, Inner, Last. These are the steps that need to be followed in order to make a quadratic equation into a simplified expression or vice versa. Each step multiplies the terms specified in the formula.

STARTING EQUATION: (X+a)(X+b)

First Terms: (X+a)(X+b)= (X)(X)= X^2

Outer Terms: (X+a)(X+b)= (X)(b)= bX

Inner Terms: (X+a)(X+b)= (a)(X)= aX

Last Terms: (X+a)(X+b)= (a)(b)= ab

End Result: X^2+bX+aX+ab

EXAMPLE 1

So, if we have the quadratic formula (X+2)(X-2), how do we factor this to get a quadratic expression like X^2+aX+bX+ab?

We first have to think about our acronym “FOIL”. Lets recall we have the equation (X+2)(X-2). 

Our first step is to multiply the First terms. The F in “FOIL” reminds us of this.  (X+2)(X-2). (X)(X)= X^2

Then, we multiply the Outer terms. (X+2)(X-2). (X)(-2)= -2X. So far, our equation is X^2-2X….

Now, following “FOIL”, we know that our Inner terms in the equation are next. So we take our equation (X+2)(X-2) and we multiply the Inner terms (X+2)(X-2). (2)(X)= 2X. Now our equation is X^2-2X+2X…..

Finally, our last step in “FOIL” is the Last terms of the equation (X+2)(X-2). So, we multiply the last terms (X+2)(X-2). (2)(-2)= -4. Now, we notice that we have a completely factored equation. Now, our equation is X^2+2X-2X-4.

But, we aren’t done yet. We have to do more simplifying because there are like terms in the equation.

2X and -2X cancel out each other because they have “X” as a like term. They also completely cancel each other out because both have “2” “X terms”. So, now the final equation is X^2-4.

X^2+2X-2X-4= X^2-4

EXAMPLE 2

If we are given the formula (X+4)(X+3), how do we factor this into a quadratic expression again?

We first “FOIL” if you remember from our first example.

So, we multiply the First terms in the equation (X+4)(X+3) together. Those terms would be both (X)(X)= X^2.

Next, if you recall we take Outer terms and multiply them as well. So, we multiply the terms in the equation (X+4)(X+3) (X)(3)= 3X. So now our equation is X^2+3X….

Now, we take our Inner terms in the equation (X+4)(X+3) and multiply 4 and X together and (4)(X)= 4X. Now, our equation is X^2+3X+4X…..

Finally, we multiply our Last terms in the equation (X+4)(X+3) together. So, that would be (4)(3)= 12. Now our equation all together is X^2+3X+4X+12

But, now we need to add or subtract like terms. We see that we can add 3X and 4X together to get 7X. That’s it. Now, our final equation is equal to X^2+7X+12.

EXAMPLE 3

If we have the same quadratic formula from EXAMPLE 2 (X^2+7X+12), how do we simplify this into the form (X+a)(X+b)?

Well, “a” and “b” have to be multiplied to equal 12 (IN THIS EXAMPLE) if you recall from the L in “FOIL”. A good trick is to find the factors of (a)(b) in the equation (X+a)(X+b), (In this example its 12) and you have to find factors that add/subtract to equal the middle term of the equation (In this equation its 7X).

First, what are the factors of 12?

Factors of 12 are…. 1 and 12, 4 and 3, and that’s it right?

Yes, that’s all the factors of 12.

(X+_)(X+_)= (X^2+7X+12)

Now, what values would add up to 7 and multiply to equal 12?

We take 4 and 3 because 4+3=7 so lets see what we get if we “FOIL” them into the equation.

(X+4)(X+3)= X^2+4X+3X+12= X^2+7X+12

F:(X)(X)= X^2

O:(X)(3)= 3X

I:(4)(X)= 4X

L:(4)(3)= 12

Sweet!