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Math Help - Factorization

  1. #1
    Junior Member mathhomework's Avatar
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    Question Factorization

    How do you factorize the following?

    1. x^4+x^2y^2+y^4


    2.(a+2b+c)^3-(a+b)^3-(b+c)^3

    3. (x+1)(x+2)+1/4
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  2. #2
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    Quote Originally Posted by mathhomework View Post
    How do you factorize the following?

    1. x^4+x^2y^2+y^4


    2.(a+2b+c)^3-(a+b)^3-(b+c)^3

    3. (x+1)(x+2)+1/4

    Quote Originally Posted by mathhomework View Post
    How do you factorize the following?

    1. x^4+x^2y^2+y^4


    2.(a+2b+c)^3-(a+b)^3-(b+c)^3

    3. (x+1)(x+2)+1/4
    1) x^4 + x^2y^2 + y^4 = x^4 + 2x^2y^2 + y^4 - x^2y^2

    = (x^2 + y^2)^2 - (xy)^2

    Now simplify.

    For (2), use x^3 + y^3 = (x+y)(x^2 -xy+y^2) Simplify the second and third term. Then proceed.

    For (3), simplify the brackets and proceed.
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  3. #3
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    Hello, mathhomework!

    1. Factor: . x^4+x^2y^2+y^4

    Add and subtract x^2y^2\!:

    . . x^4 +  x^2y^2 {\color{red}\:+\: x^2y^2} + y^4 {\color{red}\:-\: x^2y^2}

    . . . . . =\;(x^4 + 2x^2y^2 + y^4) - x^2y^2

    . . . . . =\; (x^2+ y^2)^2 - (xy)^2 \quad \leftarrow\:\text{ difference of squares}

    . . . . . =\;(x^2+y^2 - xy)(x^2+y^2+xy)




    3. Factor: . (x+1)(x+2)+\tfrac{1}{4}

    We have: . x^2 + 3x + 2 + \tfrac{1}{4} \;=\;x^2 + 3x + \tfrac{9}{4} \;=\;\left(x + \tfrac{3}{2}\right)^2



    Darn! . . . too slow ... again!
    .
    Last edited by Soroban; May 28th 2010 at 04:59 AM.
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  4. #4
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    Hello, mathhomework!

    I think I've got #2 . . .


    2.\;\;(a+2b+c)^3-(a+b)^3-(b+c)^3

    \text{We have: }\;(a+2b+c)^3 - \underbrace{\bigg[(a+b)^3 + (b+c)^3\bigg]}_{\text{sum of cubes}}

    . . . . . =\;(a+2b+c)^3 - \bigg[(a+b)(b+c)\bigg]\bigg[(a+b)^2 - (a+b)(b+c) + (b+c)^2\bigg]

    . . . . . =\; (a+2b+c)^3 - (a +2b+c)(a^2 + b^2 + c^2 + ab + bc - ac)


    \text{Factor: }\;(a+2b+c)\,\bigg[(a+2b+c)^2 - (a^2+b^2+c^2 + ab + bc - ac)\bigg]

    . . . . . =\;(a+2b+c)\left(3b^2 + 3ab + 3bc + 3ac\right)

    . . . . . =\; 3(a+2b+c)\left(b^2 + ab + bc + ac\right)

    . . . . . =\;3(a+2b+c)\bigg[b(b+a) + c(b + a)\bigg]

    . . . . . =\;3(a+2b+c)(b+a)(b+c)


    \text{Answer: }\;3(a+b)(b+c)(a + 2b + c)

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