1. ## Factorisation

Factorise :$\displaystyle 2a^4+a^2b^2+ab^3+b^4$
I could not solve this problem for factorisation. I tried a various things like grouping terms from which common factor can be taken out. Then ,as it is a degree 4 polynomial, I tried adjust terms so that the polynomial can be written as product of 4 binomials and many other things. but nothing worked out. Complex factors are not allowed. Please help me.

2. Originally Posted by amey
Factorise : $\displaystyle 2a^4+a^2b^2+ab^3+b^4$
I could not solve this problem for factorisation. I tried a various things like grouping terms from which common factor can be taken out. Then ,as it is a degree 4 polynomial, I tried adjust terms so that the polynomial can be written as product of 4 binomials and many other things. but nothing worked out.
If you divide through by $\displaystyle b^4$ and write $\displaystyle x=a/b$, then you are trying to factorise $\displaystyle 2x^4+x^2+x+1$. That polynomial has no real roots and hence no (real) linear factors. So the only hope is to factorise it as the product of two quadratics: $\displaystyle 2x^4+x^2+x+1 = (2x^2+px+q)(x^2+rx+s)$. Juggling with various possibilities for $\displaystyle p,q,r,s$, you should be able to find a factorisation.

In terms of $\displaystyle a$ and $\displaystyle b$, that will tell you that $\displaystyle 2a^4+a^2b^2+ab^3+b^4 = (2a^2+pab+qb^2)(a^2+rab+sb^2)$.

Spoiler:
$\displaystyle 2x^4+x^2+x+1 = (2x^2+2x+1)(x^2-x+1)$

3. Thanks a lot for the way out.