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Math Help - Difference of Cubes

  1. #1
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    Difference of Cubes

    I have a real hard time understading the difference of cubes.
    However, there are several questions on this topic in the coming June state exam.

    How do I play with this topic?

    QUESTIONS:

    (1) Factor a^3b^3 - 8x^6y^9

    (2) x^3 + 27
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  2. #2
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    Quote Originally Posted by symmetry View Post
    I have a real hard time understading the difference of cubes.
    However, there are several questions on this topic in the coming June state exam.

    How do I play with this topic?

    QUESTIONS:

    (1) Factor a^3b^3 - 8x^6y^9

    (2) x^3 + 27
    You have to memorize the factoring of the forms (x^3 +y^3) and (x^3 -y^3).
    x^3 +y^3 = (x+y)(x^2 -xy +y^2) -----(i)
    x^3 -y^3 = (x-y)(x^2 +xy +y^2) ------(ii)

    (1) (a^3)(b^3) -8(x^6)(y^9)
    = (ab)^3 -[2(x^2)(y^3)]^3
    = [(ab) -2(x^2)(y^3)]*{(ab)^2 +(ab)[2(x^2)(y^3)] +[2(x^2)(y^3)]^2}
    = [ab -2(x^2)(y^3)]*[(a^2)(b^2) +2ab(x^2)(y^3) +4(x^4)(y^6)] ----answer.

    (2) x^3 +27
    = x^3 +3^3
    = (x+3)(x^2 -x*3x +3^2)
    = (x+3)(x^2 -3x +9) -------------answer.
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  3. #3
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    Quote Originally Posted by symmetry View Post
    I have a real hard time understading the difference of cubes.
    However, there are several questions on this topic in the coming June state exam.

    How do I play with this topic?

    QUESTIONS:

    (1) Factor a^3b^3 - 8x^6y^9

    (2) x^3 + 27
    Ticbol says you have to memorise the factorisation of the sum and
    difference of two cubes, but in fact if you know that sums or differences
    of two cubes are involved you can deduce what the formulas are on the fly.

    Consider f(x) = x^3+y^3, if x=-y, then f(x)=0, so (x+y) is a factor of f(x).

    so we may write:

    x^3+y^3=(x+y)(x^2 + A xy + y^2) = x^3 + A x^2y + xy^2 +yx^2 + Axy^2+y^3

    Collect the terms with like powers of x and y:

    x^3+y^3=(x+y)(x^2 + A xy + y^2) = x^3 + (A+1) x^2y + (A+1)yx^2 + y^3

    So as the two sides are equal (A+1)=0, or A=-1, and:

    x^3+y^3=(x+y)(x^2 - xy + y^2).

    This could have been obtained using polynomial long division once we know
    that (x+y) is a factor of f(x).

    The other factorisation can be obtained in a similar manner by observing
    that (x-y) is a factor of x^3-y^3, or by replacing y by -y' throughout
    the factorisation already obtained.

    RonL
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  4. #4
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    Hello, symmetry!

    I assume you know the sum and difference formulas.
    . . \begin{array}{cc}A^3 + B^3 \:=\:(A + B)(A^2 - AB + B^2) \\ A^3 - B^3 \:=\:(A - B)(A^2 + AB + B^2)\end{array}

    Where is your difficulty?
    . . Understanding them?
    . . Memorizing them?

    . . . . . . . . . . . . . . . . . . . . . . . \underbrace{\text{plus or minus}}_{\downarrow}
    First, we must recognize the form: . A^3 \pm B^3
    . . . . . . . . . . . . . . . . . . . . . . . . \overbrace{\text{cube}}^{\uparrow}\;\overbrace{\te  xt{cube}}^{\uparrow}


    The cubes break up into a linear factor and a quadratic factor.

    . . . . . . \underbrace{(A \;\pm \;B)}_{\text{linear}} \underbrace{(A^2 \;\mp \;AB \;+ \;B^2)}_{\text{quadratic}}


    To memorize the signs, think of the word SOAP.

    . . A^3 \;\pm \;B^3 \:=\:(A \;\pm\;B)\;(A^2 \;\mp \;AB \;+ \;B^2)
    . . . . . . . . . . . . . . \uparrow\qquad\qquad\;\uparrow\qquad\quad\uparrow
    . . . . . . . . . . . . Same . . Opposite . Always Positive

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  5. #5
    Forum Admin topsquark's Avatar
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    Actually you don't need to memorize them, though memorization IS handy. As long as you recall that a^3 \pm b^3 is divisible by a \pm b I think this provides a good exercise for that long division you mentioned in another thread.

    -Dan
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by topsquark View Post
    Actually you don't need to memorize them, though memorization IS handy. As long as you recall that a^3 \pm b^3 is divisible by a \pm b I think this provides a good exercise for that long division you mentioned in another thread.

    -Dan
    There seems to be a bit of an echo in here today .

    RonL
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  7. #7
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    ok

    I want to thank every tutor for your insight and tips.

    Yes, my problem has been memorizing difference of cubes formulas.
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    There seems to be a bit of an echo in here today .

    RonL
    Oops! I guess it pays to read the details of each post in the thread, which I obviously didn't do here.

    -Dan
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