Question

x^4+x^2+1
can any one help me in factoring this maths and sorry for my bad english

Answer 1

x⁴+ x² + 1

x⁴+ 1 + x²

Add and subtract 2(x²)(1) to make a perfect square

= (x²)² − 2(x²)(1) + 2(x²)(1) + (1)² + x²

= (x² + 1)² − 2x² + x²

= (x² + 1)² − x²

= (x² + 1)² − (x)²

Factorize as difference of 2 squares, a² − b² = (a + b)(a − b)

= [ (x² + 1) + (x) ][ (x² + 1) − (x) ]

= (x² + 1 + x)(x² + 1 − x)

= (x² + x + 1)(x² − x + 1)

Hope this helps

Answer 2

Let x2= p

Then x4= p2

x4+ x2+ 1

= p2+ p + 1

= p2+ 1 + p

= (p)2+ (1)2+ 2*p*1 - p

= (p)2+ (1)2+ 2p - p

= (p + 1)2- p

= (p + 1)2- (√p)2

= (p + 1 + √p) (p + 1 - √p)

Substituting the value of p,

( x2+ 1 + √x2) (x2+ 1 - √x2)

= (x2+ 1 +x) (x2+ 1 - x)