# Week 5 – Precalc 11

One of the things we learned this week in Precalc 11 is Factoring Polynomials. The goal is to turn something like $3x^2+4x-32$ into (x+4)(3x-8)

To factor a polynomial, first we have to check, is there something common between the terms.
Ex. $2x^2+8x+4$
$=2(x^2+4x+2)$
In this example 2 is common between the terms, so we place the 2 infront, divide the polynomial by 2 and place them inside the brackets.

Next, we have to check if the polynomial is a difference of squares.
Ex. $25x^2-9$
$=(5x-3)(5x+3)$
We know this is a difference of squares because it’s a binomial and it’s subtracting the terms. To factor a difference of squares, find out what number squared is equal to the first term ($25x^2$) which is 5x and the last term (9). which is 3. The factored form will always be (a-b)(a+b) or (a+b)(a-b)

Next, check if there’s a pattern, if the pattern is $x^2, x,$ # then it’s an easier polynomial, however if the pattern is $x^2, x,$ # then it’s a harder one to factor. There is a few ways to factor a polynomial, for me, I like to use the grouping method.

Ex. $x^2+2x-8$
The first step in factoring an easy polynomial is to find what two numbers are multiplied to get to the last term (-8) and when added together, it adds up to the middle term (2). numbers that multiply to 8 are, 1 and 8, 2 and 4, since it’s a -8 we can assume that one number has to be negative, +4 and -2 would work so we replace the middle term with +4x and -2x.
$x^2+4x-2x-8$
Then we take what is common with the first ($x^2$) and second term (4x) which is x, then we divide both terms by x which results to x(x+4). Then we do the same thing on the third and forth term. -2(x+4). the terms in brackets should match up, if it doesn’t then you might have done something wrong, so we combine the terms in brackets and the coefficient in another brackets.
$x(x+4)-2(x+4)$
$(x+4)(x-2)$

Now let’s try a harder polynomial, it’s not really harder, we just add another step.
Ex. $3x^2+x-4$
We multiply the first number to the last, 3x-4=-12. then we find two numbers that multiply to -12, and adds up to 1, which is +4 and -3. From here on, the steps are basically the same.
$3x^2+4x-3x-4$
$x(3x+4)-1(3x+4)$
$(3x+4)(x-1)$

If the terms in a polynomial has nothing in common, or the polynomial isn’t a differrence of squares, or has no easy or hard pattern, then the polynomial is factored.