Advanced Factorization

Definition

Factorization is the decomposition of an expression into a product of its factors.

The following are common factorizations.

For any positive integer $n$, \[a^n-b^n = (a-b)(a^{n-1} + a^{n-2} b + \ldots + ab^{n-2} + b^{n-1} ).\] In particular, for $ n=2$, we have $ a^2-b^2=(a-b)(a+b)$.

For $n$ an odd positive integer, $a^n+b^n = (a+b)(a^{n-1} - a^{n-2} b + \ldots - ab^{n-2} + b^{n-1} ).$

$a^2 \pm 2ab + b^2 = (a\pm b)^2$

$x^3 + y^3 + z^3 - 3 xyz = (x+y+z) (x^2+y^2+z^2-xy-yz-zx)$

$(ax+by)^2 + (ay-bx)^2 = (a^2+b^2)(x^2+y^2)$ . $(ax-by)^2 - (ay-bx)^2 = (a^2-b^2)(x^2-y^2)$ .

$x^2 y + y^2 z + z^2 x + x^2 z + y^2 x + z^2 y +2xyz= (x+y)(y+z)(z+x)$ .

Factorization often transforms an expression into a form that is more easily manipulated algebraically, that has easily recognizable solutions, and that gives rise to clearly defined relationships.

Worked Examples

1. Find all ordered pairs of positive integer solutions $(x,y)$ such that $2^x+ 1 = y^2$ .

Solution: We have $2^x = y^2-1 = (y-1)(y+1)$ . Since the factors $(y-1)$ and $(y+1)$ on the right hand side are integers whose product is a power of 2, both $(y-1)$ and $(y+1)$ must be powers of 2. Furthermore, their difference is

$(y+1)-(y-1)=2,$

implying the factors must be $y+1 = 4$ and $y-1 = 2$ . This gives $y=3$ , and thus $x=3$ . Therefore, $(3, 3)$ is the only solution.

2. Factorize the polynomial

$f(a, b, c) = ab(a^2-b^2) + bc(b^2-c^2) + ca(c^2-a^2).$

Solution: Observe that if $a=b$ , then $f(a, a, c) =0$ ; if $b=c$ , then $f(a, b, b)=0$ ; and if $c=a$ , then $f(c,b,c)=0$ . By the Remainder-Factor Theorem, $(a-b), (b-c),$ and $(c-a)$ are factors of $f(a,b,c)$ . This allows us to factorize

$f(a,b,c) = -(a-b)(b-c)(c-a)(a+b+c).$

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Advanced Factorization

Definition

Worked Examples

1. Find all ordered pairs of positive integer solutions (x,y) (x,y)(x,y) such that 2x+1=y22^x+ 1 = y^22x+1=y2.

2. Factorize the polynomial

Comments

1. Find all ordered pairs of positive integer solutions $(x,y)$ such that $2^x+ 1 = y^2$ .