How to Factor Polynomials

Introduction

I have another page on this, which listed 10 approaches to factoring. I stopped it at #9 Long Division because it said I must understand Factor Theorem and Rational Zero Theorem which I did not. This led me in the end to here https://www.video-tutor.net/polynomials.html and why this exists on my wiki

Greatest Common Factor (GCF)

This is where you look at a polynomial and find, you guest it, the greatest common factor so given

3x + 15

It would be 3 because

3(x + 5)

And

4x² + 8x = 4x(x+2) GCF = 4x
5x² -15x³ = 5x²(1 -3x) GCF = 5x²

Factoring by Grouping

Here we divide the polynomial into two group and take out the GCF

x³ - 4x² + 3x -12

So group one

x³ - 4x² = x²(x-4)

Group two

3x -12 = 3(x - 4)

So they both have (x-4) so we can say

(x-4)(x² + 3)

Next example

  2r³ -6r² + 5r - 15
= 2r²(r -3) +5(r -3)
= (r-3)(2r² +5)

Factoring by Trinomials

For these you look for numbers which multiply to the constant and add up to the coefficient of the x term

x² + 3x - 28

So in the case two numbers which multiply to -28 and add up to +3

So 28 factors are 1, 28, 2, 14, 7, 4 - so 7 and 4

(x -4)(x+7)

And given

x² - 3x - 10

So factors of 10 are 1,10,2,5 - so 5 and 2

(x-5)(x+2)

When the first term is not x² for example

2x² + 20x + 48

Just get rid of the 2

2(x² + 10x + 24)
2(x+4)(x+6)

What do you do I you cannot factor out to x². A tiny bit more complex this bit because the slight of hand of the youtuber. You need to multiply the coefficient of the first term with the constant and then look for two numbers which when added together = the middle term. So given

2x² - 5x + -3

So 2x -3 = -6 and factors of -6 are 2, -3, 1 -6
So 1 -6 = -5 = the middle term coefficient<br And this was the tricky bit. Just replace p-5x with -6x +1x. It is obvious when you see it but they mean replace the middle term with the factors you identified. In this case 1 and -6. So we have

2x² - 6x + 1x -3

So using the grouping approach take out the GCF we have

2x(x-3) + 1(x-3)
(x-3)(2x+1)