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Math Help - please help!

  1. #1
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    please help!

    1. factor the expression x^3 + 8y^3
    2. factor x^9 - 27y^6
    3. factor 4x^2-9
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  2. #2
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    Solutions

    1. factor the expression x^3 + 8y^3
    2. factor x^9 - 27y^6
    3. factor 4x^2-9

    1.Well, since the first one is the sum of two cubes, you write it of the form:
    (a + b)(a^2 - ab + b^2).

    The cube root of x^3 is x because x*x*x = x^3.
    The cube root of 8y^3 is found by taking the cube root of 8 and the cube root of y^3 and multiplying them together.

    So, cube root of 8 is 2 and cube root of y^3 is y. 2y*2y*2y = 8y^3.

    As per the above terms, we should find a^2, ab and b^2.
    Allow a=x and b=2y

    a^2 = x*x = x^2
    b^2 = 2y*2y = 4y^2
    ab = x*2y = 2xy

    Now, we just use substitution:
    (x + 2y)(x^2 - 2xy + 4y^2). and we are done.
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  3. #3
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    Solution 2

    1. factor the expression x^3 + 8y^3
    2. factor x^9 - 27y^6
    3. factor 4x^2-9

    1.Well, since the second one is the difference of two cubes, you write it of the form:
    (a - b)(a^2 + ab + b^2).

    The cube root of x^9 is x^3 because x^3*x^3*x^3 = x^9. You add the exponents when the terms are multiplied.

    The cube root of 27y^6 is found by taking the cube root of 27 and the cube root of y^6 and multiplying them together.

    So, cube root of 27 is 3 and cube root of y^6 is y^2. 3y^2*3y^2*3y^2 = 27y^6. The exponents are added, the base remains the same, and the coefficients are multiplied.

    As per the above terms, we should find a^2, ab and b^2.
    Allow a=x^3 and b=3y^2

    a^2 = x^3*x^3 = x^6
    b^2 = 3y^2*3y^2 = 9y^4
    ab = x^3*3y^2 = 3(x^3)(y^2)

    Now, we just use substitution:
    (x^3 - 3y^2)(x^6 + 3(x^3)(y^2) + 9y^4). and we are done.
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  4. #4
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    Solution 3

    3. factor 4x^2-9

    This is the difference of squares, so it is of the form: (a + b)(a - b)

    Take the square root of 4x^2 to find a and the square root of 9 to find b

    square root of 4 is 2 and square root of x^2 is x
    So, 2x is the square root of 4x^2 --> 2x*2x = 4x^2

    square root of 9 is 3 --> 3*3 = 9

    Let a = 2x and b = 3

    Now just substitute:
    (2x + 3)(2x - 3), and we are done
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