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One of the things I've struggled with most in algebra/calculus is all the "factoring tricks". When I take time away from doing math I inevitably forget most if not all of them. The old proverb "use it or lose it" definitely comes into play.

I'm a huge fan of Khan Academy and have started to keep a list of tutorials which demonstrate factoring, but I haven't found them all:

Was wondering if anyone could help me fill in the gaps by providing their own list of factoring tips be them in the thread itself, or to online resources like Khan Academy, this site, or elsewhere

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    $\begingroup$ It is not possible to answer your question without reading through all your links and lessons, because who knows what is covered in "Factoring Special Products". So it would be fine to indicate shortly what kind of expressions are addressed. $\endgroup$
    – Phira
    May 10, 2011 at 18:28
  • $\begingroup$ Also you have to be aware that you are also asking about applications of factoring methods in your examples. $\endgroup$
    – Phira
    May 10, 2011 at 18:36
  • $\begingroup$ "flagged" for community wiki. $\endgroup$
    – Rasmus
    May 10, 2011 at 20:22
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    $\begingroup$ It seems that you are interested in factoring polynomial and not in factoring integers. But you do not mention this in your question. $\endgroup$
    – miracle173
    Jan 14, 2021 at 21:52

6 Answers 6

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Remembering the geometric series helps for factoring things like $a^n-b^n$:

$$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})$$

A similar factorization applies for $a^\ell+b^\ell$ for $\ell$ odd:

$$\begin{align*} a^3+b^3&=(a+b)(a^2-ab+b^2)\\ a^5+b^5&=(a+b)(a^4-a^3 b+a^2 b^2-ab^3+b^4) \end{align*}$$

whose general pattern I'll leave for you to tease out.

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Completing a square with a middle term that is a square itself:

$$x^4+4=x^4+4+4x^2-4x^2=(x^2+2)^2-(2x)^2=(x^2+2-2x)(x^2+2+2x).$$

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    $\begingroup$ Also $x^4-x^2+16=x^4+8x^2+16-9x^2=(x^2+4)^2-(3x)^2=(x^2+4-3x)(x^2+4+3x)$. $\endgroup$
    – Isaac
    May 10, 2011 at 19:59
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Completing the rectangle:

$$ab+7a+11b=ab+7a+11b+77-77=(a+11)(b+7)-77.$$

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As indicated in my comment above, you did not really say what material is covered in your links, but in any case I suggest you look at the algebra section of alcumus:

http://www.artofproblemsolving.com/Alcumus/Introduction.php

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Consider also what trinomials look like when squared:

$(a+b+c)^2=a^2+b^2+c^2+2(ab+ac+bc)$

This allows you to take the general two variable quadratic to a linear function of two squares with integer coefficients $(a,b,c,d,e,f)\in \mathbb{Z}^6$:

$ax^2+bxy+cy^2+dx+ey=f \qquad \to$

$\bigg[(4ac-b^2)y+2ae-bd\bigg]+(4ac-b^2)\bigg[2ax+by+d\bigg]^2=(4ac-b^2)(4af+d^2)+(2ae-bd)^2$

Or if you let $$ D=4ac-b^2$$ $$f(x,y)=2ax+by+d$$ $$g(y)=Dy+2ae-bd$$ $$A=D\cdot(4af+d^2)+(2ae-bd)^2 \in \mathbb{Z}$$

it looks like

$\bigg[g(y)\bigg]^2+D\cdot \bigg[f(x,y)\bigg]^2=A$

The above can help when you study diophantine equations

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There are probably 30 different ways to factor. Don't get hung up like me about trying to find every way to factor. You don't have to be Ash and "Catch them all". Otherwise, you're going to be Alice in Mathland when really, you just need to move on and learn harder maths. I'm sure I'm not the only one who has other things to do in their day. I was taught the X box method. The pro is that it's very versatile, the con is it's a little bit excessive for simple factoring.

X box factoring example: https://www.youtube.com/watch?v=Sx4NvCMtlj0

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