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Math Help - Factorisation

  1. #1
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    Factorisation

    Factorise : 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.
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  2. #2
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    Quote Originally Posted by amey View Post
    Factorise : 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 b^4 and write x=a/b, then you are trying to factorise 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: 2x^4+x^2+x+1 = (2x^2+px+q)(x^2+rx+s). Juggling with various possibilities for p,q,r,s, you should be able to find a factorisation.

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

    Spoiler:
    2x^4+x^2+x+1 = (2x^2+2x+1)(x^2-x+1)
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  3. #3
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    Thanks a lot for the way out.
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