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Basic Factoring (Posted on 2007-03-21) Difficulty: 2 of 5
When math students are taught factoring, usually the first thing they learn are special factoring identities such as:

1: a-b= (a-b)(a+b)
2: a-b = (a-b)(a+ab+b)
3: a+b = (a+b)(a-ab+b)

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

See The Solution Submitted by Brian Smith    
Rating: 3.0000 (1 votes)

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Solution An Attempt To The Solution | Comment 2 of 4 |

Let us substitute x^2 = p

Then:

x^4 + x^2 + 1

= p^2 + p + 1

= p^2 + 2p + 1 - p

= (q-1)^2 + 2(q-1) + 1 - p [substituting q = p+1, so that p = q-1]

= (q-1)(q-1) + 2(q-1) + 1 - x^2 [resubstituting the value of p]

= q(q-1) - (q-1) + 2(q-1) + 1 - x^2

= q(q-1) + (q-1) + 1 - x^2

= (q-1)(q+1) + 1 - x^2

= (q^2 - 1) + 1 - x^2 [By Identity (i)]

= q^2 - x^2

= (q-x)(q+x) [By Identity (i)]

= (p-x+1)(p+x+1) [Resubstituting the value of q]

= (x^2-x+1)(x^2+x+1) [Resubstituting the value of p in the expression]

Edited on March 22, 2007, 3:57 am
  Posted by K Sengupta on 2007-03-22 00:42:24

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