perplexus.info :: Just Math : Basic Factoring

Basic Factoring (Posted on 2007-03-21)

When math students are taught factoring, usually the first thing they learn are special factoring identities such as:

1: a²-b²= (a-b)(a+b)
2: a³-b³ = (a-b)(a²+ab+b²)
3: a³+b³ = (a+b)(a²-ab+b²)

Find a factorization of the expression x⁴+x²+1 using only those identities.

Submitted by Brian Smith

Rating: 3.0000 (1 votes)

Solution: (Hide)

Start with:
x^6-1 = x^6-1
Apply (1) to the left side and (2) to the right side:
(x^3-1)(x^3+1) = (x^2-1)(x^4+x^2+1)
Next, apply (2) and (3) to the respective factors on the left side and apply (1) to (x^2-1) on the right side
(x-1)(x^2+x+1)(x+1)(x^2-x+1) = (x-1)(x+1)(x^4+x^2+1)
Since (x-1)(x+1) appears on each side, the factorization of x^4+x^2+1 is (x^2+x+1)(x^2-x+1)
Check the comments for more ways to solve the problem, some of which include other equally basic factoring identites and methods.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Another means of solution	Mike C	2007-03-22 14:32:29
solution	xdog	2007-03-22 09:53:51
An Attempt To The Solution	K Sengupta	2007-03-22 00:42:24
How's this? (Probable spoiler)	Jer	2007-03-21 11:07:37

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