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Math Help - Factoring and simplifying an expression

  1. #1
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    Factoring and simplifying an expression

    Hi, the problem is to factor and simplify the expression

    (x^2-4)(x^2+3)^{1/2} - (x^2-4)^2(x^2+3)^{3/2}

    I can factor the left side but the right leaves me stuck.

    I tried multiplying both terms in the right and got (x^4-8x^2+16) \sqrt{x^6+9x^4+27x^2+27}
    but I feel like I am going the wrong way and don't think it should be this complicated.

    The answer in the back of the book is (x+2)(x-2)(x^2+3)^{1/2}(-x^4 +x^2 +13)
    I don't know where the final term came from.

    Thank you.
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  2. #2
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    Re: Factoring and simplifying an expression

    Quote Originally Posted by Foxlion View Post
    Hi, the problem is to factor and simplify the expression

    (x^2-4)(x^2+3)^{1/2} - (x^2-4)^2(x^2+3)^{3/2}

    I can factor the left side but the right leaves me stuck.

    I tried multiplying both terms in the right and got (x^4-8x^2+16) \sqrt{x^6+9x^4+27x^2+27}
    but I feel like I am going the wrong way and don't think it should be this complicated.

    The answer in the back of the book is (x+2)(x-2)(x^2+3)^{1/2}(-x^4 +x^2 +13)
    I don't know where the final term came from. Thank you.
    (x^2 - 4)(x^2 + 3)^{\frac{1}{2}}[1 - (x^2 - 4)(x^2 + 3)]

    (x + 2)(x - 2) \sqrt{x^2 + 3} \ [1 - x^4 - 3x^2 + 4x^2 + 12]

    Now just combine alike terms in the brackets.

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  3. #3
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    Re: Factoring and simplifying an expression

    where does the 1 come from?
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  4. #4
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    Re: Factoring and simplifying an expression

    Quote Originally Posted by Foxlion View Post
    where does the 1 come from?
    [{\color{blue}1} - (x^2 - 4)(x^2 + 3)]=[{\color{blue}1} - (x^4-x^2-12)]
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  5. #5
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    Re: Factoring and simplifying an expression

    Hello, Foxlion!

    \text{Factor and simplify: }\:(x^2-4)(x^2+3)^{\frac{1}{2}} - (x^2-4)^2(x^2+3)^{\frac{3}{2}}

    \text{Book answer: }\:(x+2)(x-2)(x^2+3)^{\frac{1}{2}}(-x^4 +x^2 +13)

    \text{If you had: }\,ab^{\frac{1}{2}} - a^2b^{\frac{3}{2}},\:\text{ can you see that:}
    . . \text{both terms have a factor of }a,
    . . \text{both terms have a factor of }b^{\frac{1}{2}}\,?

    \text{Factor out }ab^{\frac{1}{2}},\:\text{ and we have: }\:ab^{\frac{1}{2}}(1 - ab)


    If you can't follow that, you need more help than we can provide.
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  6. #6
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    Re: Factoring and simplifying an expression

    no I get that, but how does

    (x^2-4)^2(x^2+3)^{3/2}

    become  1-(x^2-4)(x^2+3) ?
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  7. #7
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    Re: Factoring and simplifying an expression

    yeah I see that, thank you.
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  8. #8
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    Re: Factoring and simplifying an expression

    Hello again, Foxlion!

    \text{Factor and simplify: }\:(x^2-4)(x^2+3)^{\frac{1}{2}} - (x^2-4)^2(x^2+3)^{\frac{3}{2}}

    Okay, you understand this: . ab^{\frac{1}{2}} - a^2b^{\frac{3}{2}} \:=\:ab^{\frac{1}{2}}(1 - ab) .[1]

    The given problem is a messier version of this equation.
    . . They used (x^2-4) instead of a
    . . and used (x^2+3) instead of b.

    Substitute those expression into [1].

    Got it?
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  9. #9
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    Re: Factoring and simplifying an expression

    Quote Originally Posted by soroban View Post
    hello again, foxlion!


    okay, you understand this: . ab^{\frac{1}{2}} - a^2b^{\frac{3}{2}} \:=\:ab^{\frac{1}{2}}(1 - ab) .[1]

    the given problem is a messier version of this equation.
    . . they used (x^2-4) instead of a
    . . and used (x^2+3) instead of b.

    substitute those expression into [1].

    Got it?


    got it
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