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Math Help - Factorization, Rationalization...

  1. #1
    Newbie Calculus is Amazing's Avatar
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    Factorization, Rationalization...

    While studying Calculus, often algebraic methods are used to factorize, simplify, or rationalize (I think) expressions, and those algebraic methods are not explained, because the person who made the book/video assumed that if someone was learning calculus they new algebraic methods. Well, could you guys please give me an explanation of factorization and rationalization so my memory is refreshed? And could someone please remind me of all the methods of simplification?
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  2. #2
    Super Member craig's Avatar
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    Factorisation involves finding and 'taking' out factors of equations.

    For example, if we have the equation x^2 - 4x = 0 we can see that both parts to this equation have a common factor of x, we can therefore write this as:

    x(x - 4) = 0.

    As you will know from very early maths, if two things multiplied together equal 0, then one of them is zero.

    Using this fact we can then state that either

    x = 0 or x - 4 = 0

    Therefore x = 0 \text{ or } 4.

    This can also be applied to quadratics, a quadratic equation such as

    x^2 - 4x + 3 = 0 can be factorised to (x - 3)(x - 1)

    This is a slightly different method, but there is a good article on factorising quadratics here.
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  3. #3
    Super Member craig's Avatar
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    Just been trying to think of an example of rationalising and the only thing I could think of is rationalising the denominator or a fraction containing a surd.

    As you may or may not know, a surd is a number such as \sqrt{2} that cannot be express as a 'normal' number, ie. you can't write it as a finite decimal.

    Fractions containing surds as a denominator aren't considered to be 'elegent' as my maths tutor once put it. For example:

    \frac{3}{\sqrt{2}} can be rationalised to become \frac{3\sqrt{2}}{2} by multiplying by \frac{\sqrt{2}}{\sqrt{2}}, this is considered rationalised.

    Another example is where you have a fraction such as \frac{3}{\sqrt{2} + \sqrt{5}}.

    To deal with this you use the Difference of two squares identity, multiplying top and bottom by (\sqrt{2} - \sqrt{5}).

    This gives you \frac{3(\sqrt{2} - \sqrt{5})}{(\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5})}

    \frac{3(\sqrt{2} - \sqrt{5})}{\sqrt{2}^2 + \sqrt{10} -\sqrt{10} - \sqrt{5}^2}

    \frac{3(\sqrt{2} - \sqrt{5})}{-3}

    \sqrt{5} - \sqrt{2}

    Hope it was something like this that you were looking for, if there is anything more that you don't understand please post back
    Last edited by craig; June 3rd 2009 at 12:21 AM.
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  4. #4
    Newbie Calculus is Amazing's Avatar
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    Quote Originally Posted by craig View Post
    Just been trying to think of an example of rationalising and the only thing I could think of is rationalising the denominator or a fraction containing a surd.

    As you may or may not know, a surd is a number such as \sqrt{2} that cannot be express as a 'normal' number, ie. you can't write it as a finite decimal.

    Fractions containing surds as a denominator aren't considered to be 'elegent' as my maths tutor once put it. For example:

    \frac{3}{\sqrt{2}} can be rationalised to become \frac{3\sqrt{2}}{2} by multiplying by \sqrt{2}, this is considered rationalised.

    Another example is where you have a fraction such as \frac{3}{\sqrt{2} + \sqrt{5}}.

    To deal with this you use the Difference of two squares identity, multiplying top and bottom by (\sqrt{2} - \sqrt{5}).

    This gives you \frac{3(\sqrt{2} - \sqrt{5})}{(\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5})}

    \frac{3(\sqrt{2} - \sqrt{5})}{\sqrt{2}^2 + \sqrt{10} -\sqrt{10} - \sqrt{5}^2}

    \frac{3(\sqrt{2} - \sqrt{5})}{-3}

    \sqrt{5} - \sqrt{2}

    Hope it was something like this that you were looking for, if there is anything more that you don't understand please post back
    Quote Originally Posted by craig View Post
    Factorisation involves finding and 'taking' out factors of equations.

    For example, if we have the equation x^2 - 4x = 0 we can see that both parts to this equation have a common factor of x, we can therefore write this as:

    x(x - 4) = 0.

    As you will know from very early maths, if two things multiplied together equal 0, then one of them is zero.

    Using this fact we can then state that either

    x = 0 or x - 4 = 0

    Therefore x = 0 \text{ or } 4.

    This can also be applied to quadratics, a quadratic equation such as

    x^2 - 4x + 3 = 0 can be factorised to (x - 3)(x - 1)

    This is a slightly different method, but there is a good article on factorising quadratics here.
    Thank you very much!

    But in your rationalization example #1, how does multiplying the numerator and denominator by the \sqrt{2} give the answer that you specified?
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  5. #5
    Super Member craig's Avatar
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    Quote Originally Posted by Calculus is Amazing View Post
    Thank you very much!

    But in your rationalization example #1, how does multiplying the numerator and denominator by the \sqrt{2} give the answer that you specified?

    \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}

    Becomes

    \frac{3 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}

    Which becomes

    \frac{3 \sqrt{2}}{2}

    Is this clearer now? It is considered rationalised because there is no surd on the denominator.
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