Question

factorize: a^4+a^2+1

Answer 1

Answer:

There are multiple ways to approach this problem.

Approach # 1:

[math]x^4 + x^2 + 1 = x^4 + x^3 + x^2 - x^3 - x^2 - x + x^2 + x + 1 [/math]

[math]=> x^2(x^2+ x + 1) -x(x^2 + x + 1) + (x^2+x+1)[/math]

[math]=> (x^2 + x + 1) * ( x^2 - x + 1)[/math]

Approach # 2:

Let [math]y = x^2[/math]

[math]=> x^4 + x^2 + 1 = y^2 + y + 1[/math]

Let us find the roots

[math] y^2 + y + 1 = 0[/math]

[math]=> y = \frac{-1 \pm \sqrt{1–4}}{2}[/math]

[math]=> y = \frac{1}{2} * (-1 \pm \sqrt{3}i)[/math]

[math]=> y = x^2 = \frac{1}{2} * (-1 \pm \sqrt{3}i) [/math]

[math]x^2 = [\frac{1}{2} * ( -1 \pm \sqrt{3}i) ]^2[/math]

[math]=> x = \pm [\frac{1}{2} * (-1 \pm \sqrt{3}i) ][/math]

The 4 roots are

[math]R_1 = + [\frac{1}{2} * (-1 + \sqrt{3}i) ][/math]

[math]R_2 = + [\frac{1}{2} * (-1 - \sqrt{3}i) ][/math]

[math]R_3 = - [\frac{1}{2} * (-1 + \sqrt{3}i) ][/math]

[math]R_4 = - [\frac{1}{2} * (-1 - \sqrt{3}i) ][/math]

[math](x- R_1), (x - R_2), (x- R_3), (x - R_4)[/math] are the factors

Multipliying the factors 1 and 4 and 2 and 3 will provide the same factors as in Approach # 1.