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Thread: Rationalizing the numerator and complex factoring

  1. #1
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    Rationalizing the numerator and complex factoring

    1. [ 1 / sqroot(x)] - 1 / (x-1)
    2. [x*sqroot(x)] -8 / (x-4)

    Factor:
    a) x^6+8
    b)x^4-16
    c)r^8-1
    d) x^3-x^2-16x+16
    f)x^4-3x^3-7x^2+27x-18



    are there any easy ways to factor theses without graphing, algebraically. I only know of synthetic division.
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  2. #2
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    Re: Rationalizing the numerator and complex factoring

    There are certain formulas which one must learn. (a^3+b^3) = (a+b)(a^2-ab+b^2); (a^3-b^3) = (a+b)(a^2+ab+b^2); a^2-b^2 = (a+b)(a-b); in case of cubic and bi-quadratic polynomials we can find linear factor by checking the sum of the coefficients, if the sum of the coefficients is zero then x = 1 is a solution and (x-1) a factor. Lastly if the sum of the coefficients of the even powers of the variable is equal to the sum of the coefficients of the odd powers of the variables then x = -1 is a solution and (x+1) is a factor.
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  3. #3
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    Re: Rationalizing the numerator and complex factoring

    Hello, skg94!

    You are expected to know basic factoring and these three formulas:

    . . Difference of squares: .$\displaystyle a^2 - b^2 \:=\:(a-b)(a+b)$

    . . Sum/difference of cubes: .$\displaystyle a^3 \pm b^3 \:=\:(a\pm b)(a^2 \mp ab + b^2)$


    $\displaystyle [1]\;\frac{\frac{1}{\sqrt{x}} - 1}{x-1}$

    $\displaystyle \frac{\frac{1}{\sqrt{x}} - 1}{x-1} \;=\;\frac{\frac{1-\sqrt{x}}{\sqrt{x}}}{x-1} \;=\; \frac{1-\sqrt{x}}{\sqrt{x}(x-1)} \;=\;-\frac{\sqrt{x}-1}{\sqrt{x}(x-1)} $


    Here's where it gets sneaky . . . $\displaystyle x-1$ is a difference of squares.

    . . $\displaystyle x - 1 \;=\;(\sqrt{x})^2 - 1^2 \;=\;(\sqrt{x} - 1)(\sqrt{x} + 1)$


    The fraction becomes: .$\displaystyle -\frac{\sqrt{x}-1}{\sqrt{x}(\sqrt{x}-1)(\sqrt{x}+1)} \;=\;-\frac{1}{\sqrt{x}(\sqrt{x}+1)} $




    $\displaystyle [2]\; \frac{x\sqrt{x} - 8}{x-4}$

    We have: .$\displaystyle \frac{\overbrace{(\sqrt{x})^3 - 2^3}^{\text{diff.cubes}}}{\underbrace{(\sqrt{x})^2 -2^2}_{\text{diff.squares}}} \;=\;\frac{(\sqrt{x}-2)(x + 2\sqrt{x}+4)}{(\sqrt{x}-2)(\sqrt{x}+2)} \;=\;\frac{x+2\sqrt{x}+4}{\sqrt{x}+2}$




    $\displaystyle (a)\;x^6+8$ . sum of cubes

    $\displaystyle (x^2)^3 + 2^3 \;=\;(x^2+2)(x^4 - 2x^2 + 4)$




    $\displaystyle (b)\;x^4-16$ . diff. of squares

    $\displaystyle (x^2)^2 - 4^2 \;=\;\overbrace{(x^2 - 4)}^{{\color{blue}\text{diff.squares}}}(x^2+4) \;=\;(x-2)(x+2)(x^2+4)$




    $\displaystyle (c)\;r^8-1$ .diff. of squares

    $\displaystyle (r^4)^2 - 1^2 \;=\;\overbrace{(r^4-1)}^{{\color{blue}\text{diff.squares}}}(r^4+1) \;=\;\overbrace{(r^2-1)}^{{\color{blue}\text{diff.squares}}}(r^2+1)(r^4 +1)$

    . . . . . . . . . . $\displaystyle =\;(r-1)(r+1)(r^2+1)(r^4+1)$




    $\displaystyle (d)\;x^3-x^2-16x+16$

    $\displaystyle x^3-x^2-16x + 16 \;=\;x^2(x-1) - 16(x-1)$

    . . . . . . $\displaystyle =\;(x-1)(x^2-16) \;=\;(x-1)(x-4)(x+4)$




    $\displaystyle (e)\;x^4-3x^3-7x^2+27x-18$

    We have: .$\displaystyle x^4 - 3x^3 - 7x^2 + 27x -18$

    . . $\displaystyle =\; x^4-3x^3 + \overbrace{2x^2 - 9x^2} + 27x - 18 $

    . . $\displaystyle =\;x^2(x^2-3x+2) - 9(x^2-3x+2) $

    . . $\displaystyle =\;(x^2-3x+2)(x^2-9)$

    . . $\displaystyle =\'(x-1)(x-2)(x-3)(x+3)$
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