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Math Help - How to factor with multiple exponents?

  1. #1
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    How to factor with multiple exponents?

    I'm kind of struggling with factoring when encountering multiple exponents. Can someone please explain the steps to factor the following?

    4x^4+12x^3-40x^2
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  2. #2
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    You need to find a common factor - when there are multiple exponents on the same base then the lowest exponent is a factor of all of them - in this case x^2 is a factor (and so is 4). In this case you will get a quadratic once you take out the factors - you should check if it factorises to (which it does in this case)

    My answer is in a spoiler below

    Spoiler:
    4x^2(x+5)(x-2)
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  3. #3
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    I don't get it. Can you please break it down step by step as to how you got the 5 and -2?
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  4. #4
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    Do you understand that you can factor 4x^2 initally?

    This gives: 4x^2(x^2+3x-10)

    However, the x^2+3x+10 can also be factorised to give the answer of 4x^2(x+5)(x-2)
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  5. #5
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    Ah, ok, I got it. But what about when there is no exponent or x on the last number, such as:

    <br />
x^4-20x^2+64

    I know it is:

    (x+2)(x-2)(x+4)(x-4)

    But I want to know a simple way of break it down step by step to arrive at that. That is what I'm struggling with. Thanks.
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  6. #6
    A riddle wrapped in an enigma
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    Quote Originally Posted by softwareguy View Post
    Ah, ok, I got it. But what about when there is no exponent or x on the last number, such as:

    <br />
x^4-20x^2+64

    I know it is:

    (x+2)(x-2)(x+4)(x-4)

    But I want to know a simple way of break it down step by step to arrive at that. That is what I'm struggling with. Thanks.
    Hi softwareguy,

    x^4-20x^2+64=(x^2-16)(x^2-4)

    Now, all you have to do is factor the two "difference of squares" to reach your desired result.


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  7. #7
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    Thanks, that helps.

    One other thing. Is there any "trick" to more quickly figuring this out? For example, lets say you have a larger last number, such as:

    x^4-61x^2+900

    I'm struggling how to come up with the answer for this one.
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  8. #8
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    That is a tough one, (-30)^2=900 but that can't be right because -30 -30 \neq -61 mind you it will be close so I will try 25 which is also a factor of 900:

    900 = -25 \cdot -36. Now to check if -25-36=-61 which works.

    Hence x^4-61x^2+900=(x^2-25)(x^2-36)

    Yet these are both the difference of two squares: x^4-61x^2+900 = (x-5)(x+5)(x-6)(x+6)
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