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.

Let’s start with an easier polynomial.
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.

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