FACTORING IS FUN!
Wait...what is factoring?
Basically, factoring is determining what you need to multiply in order to get a certain expression.
Factoring helps us find the roots of an equation (when y = 0)
We already know that numbers have factors...
For example: 2 x 3 = 6
Therefore, 2 and 3 are factors of 6.
Did you know that expressions can have factors, too?!
In the expression 2y + 6, both 2y and 6 have a factor of 2.
Therefore, 2y + 6 in factored form is 2(y+3).
If we were to transform 2(y+3) into 2y + 6, that would be called expanding, which is the opposite of factoring.
NOTE: It is important to find the greatest common factor if you want to completely factor the expression.
Factoring Methods
1. Factoring Quadratics
In order to factor a quadratic equation, it needs to be in standard form.
ax2+ bx + c = 0
Once it is in standard form, for example...
2x2 + 7x + 3 = 0
Our first step is to find 2 numbers that multiply to give ac (in this case 2 x 3) and add to give b (in this case, 7).
To do this, it helps to think of or list the factors of ac (6) and try adding some to get b (7).
Once you find these numbers, they will be 1 and 6, rewrite the middle of the equation with them.
2x2 + 6x + x + 3 = 0
The first two terms factor to 2x(x+3) and (x+3). Since (x+3) is present in both factors, the factored form of this equation is...
(2x + 1)(x+3) = 0
To find the roots of this factored equation, make each set of parentheses equal to 0.
2x + 1 = 0 x + 3 = 0
x = -1/2 x = -3
FUN FACT:
*On a graph of quadratic equations, the roots are the x-intercepts of the parabola!*
2. Quadratic Formula
When an equation cannot be easily factored and therefore has irrational roots, we have to use the quadratic formula.
This formula is very helpful, and you should try to memorize it! Here's a fun song:
Let's try to factor an equation using the quadratic formula we just memorized...
3x2 – 10x + 5 = 0
3. Factoring by Grouping
Grouping is useful when there is no greatest common factor for all of the terms in an equation (usually an even number of terms). For example...
2xy + 5x + 10y + 25
Our first step is to group the first two and last two terms together, and find their respective greatest common factors.
2xy + 5x has a greatest common factor of x.
10y + 25 has a greatest common factor of 5.
x(2y + 5) + 5(2y + 5)
WOW! It looks like both groups have a common factor of 2y + 5!
Therefore this expression can be simplified into its complete factored form...
(2y + 5)(x + 5)
It is important to master factoring because it comes up in many different types of math. Knowing the different techniques will make it much easier for you to understand more complicated problems. Here are some identities to memorize as well that will help you recognize the easiest way to factor certain expressions and equations.
a2 – b2 = (a+b)(a-b)
a2 + 2ab + b2 = (a+b)(a+b)
a2 – 2ab + b2 = (a-b)(a-b)
a3 + b3 = (a+b)(a2-ab+b2)
a3 - b3 = (a-b)(a2+ab+b2)
:)
I like it that you showed all the steps. Great job.
ReplyDeleteEmily, your blog is the BEST by far! I like how you proved that factoring is fun, and how the quadratic formula can be memorized by a very cute song showing simple tools like the youtube video. Well done!!
ReplyDeleteemily,
ReplyDeletefactoring IS fun! yay! i love that little video that you included. the characters are adorbs and the song is catchy! a real life example would have been a great addition but otherwise wonderful job.
professor little