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Math Help - Factorisation Help

  1. #1
    Newbie sabyasachi's Avatar
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    Angry Factorisation Help

    PLEASE HELP!!
    1. FACTORISE x2 - bx- a2 -5ab -6b2
    2. a/b + b/a =1 , find the value of a3 + b3
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  2. #2
    Moo
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    Hello,

    Quote Originally Posted by sabyasachi View Post
    PLEASE HELP!!
    1. FACTORISE x2 - bx- a2 -5ab -6b2
    For the first one :
    Try to get quadratic forms.

    x^2-bx-a^2-5ab-6b^2

    ={\color{red}x}^2-2\cdot {\color{blue}\frac b2} \cdot {\color{red}x}-a^2-5ab-6b^2

    ={\color{red}x}^2-2\cdot {\color{blue}\frac b2} \cdot {\color{red}x}+\underbrace{\left({\color{blue} \frac b2}\right)^2-\left(\frac b2\right)^2}_{=0}-a^2-5ab-6b^2

    =\left({\color{red} x}-{\color{blue} \frac b2}\right)^2-\frac{b^2}{4}-a^2-5ab-6b^2

    =\left(x-\frac b2\right)^2-\left(a^2+5ab+\frac{25}{4} \cdot b^2\right)

    =\left(x-\frac b2\right)^2-\left({\color{red}a}^2+2 \cdot {\color{red}a} \cdot {\color{blue}\frac{5 \cdot b}{2}}+\left({\color{blue} \frac 52 \cdot b}\right)^2\right)

    =\left(x-\frac b2\right)^2-\left({\color{red}a}+{\color{blue} \frac 52 \cdot b}\right)^2

    ~~~~~~~~~~~~

    Now, you have something like x-y, which is (x-y)(x+y).
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sabyasachi View Post
    FACTORISE x2 - bx- a2 -5ab -6b2
    There is probably a nice little trick to this. But I don't see it.

    So let's do it the hard way.

    If this is going to factor it will be of the general form:
    (x + c \cdot a + d \cdot b)(x + e \cdot a + f \cdot b)

    When we expand this we get
    x^2 + ce \cdot a^2 + df \cdot b^2 + (c + e) \cdot ax + (f + d) \cdot bx + (cf + de) \cdot ab

    Comparing this to your expression we get
    ce = -1
    df = -6
    c + e = 0
    f + d = -1
    cf + de = -5

    The solution to this system is fairly simple to derive and gives c = 1, d = 2, e = -1, and f = -3.

    So
    x^2 - bx- a^2 - 5ab - 6b^2 = (x + a + 2b)(x - a - 3b)

    -Dan
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by sabyasachi View Post
    a/b + b/a =1 , find the value of a3 + b3
    \frac{a}{b} + \frac{b}{a} = 1

    Multiply both sides by ab:
    a^2 + b^2 = ab

    a^2 - ab + b^2 = 0

    Now,
    a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (a + b)(0) = 0

    -Dan

    (I see I was faster than Moo this time. )
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  5. #5
    Moo
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    (I see I was faster than Moo this time. )
    Actually, I was thinking on it, but I couldn't remember the development of a^3+b^3 ^^

    And I prefer my method to yours for the first one Factorisation Help-haha.gif.

    See you next time for another challenge ! xD

    (how are you diapers ? )
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Moo View Post
    (how are you diapers ? )
    You know, out of context that question is just sooooo wrong!

    -Dan
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  7. #7
    Moo
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    Quote Originally Posted by topsquark View Post
    You know, out of context that question is just sooooo wrong!

    -Dan
    Why ? We both know the context
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