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Thread: Greatest Common factor from polynomial: 10x​^2 ​​+ 25x + 15

  1. #1
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    Greatest Common factor from polynomial: 10x​^2 ​​+ 25x + 15

    I have no idea what I am getting wrong here...

    here is the problem:

    3) Factor out the greatest common factor in the following polynomial.

    10x​2​​+25x+15

    I tried to do two things:

    5(2x^2+1)(x+2) and
    5[(2x^2+1)(x+2)]

    both are wrong, why?

  2. #2
    Elite Member mmm4444bot's Avatar
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    Quote Originally Posted by Quick View Post
    Factor out the greatest common factor in the following polynomial.

    10x^​2 ​​+ 25x + 15
    Good morning. The instruction to "factor out the greatest common factor" of a math expression is not the same as "factor the expression".

    To factor out the greatest common factor in this quadratic polynomial means to look for the largest common factor of the coefficients (A,B,C).

    The coefficients are 10, 25, 15. The greatest common factor of this set is 5. Hence, factoring out the greatest common factor (5) results in this:

    5(2x^2 + 5x + 3)

    Done!


    If you want to continue factoring, you can now work with the easier expression, inside the parentheses. If you use separate scratch paper or work off to the side, don't forget to add that factor of 5 back in, when you're finished factoring 2x^2+5x+3 (a common mistake students make).


    5(2x^2 + 1)(x + 2)

    5[(2x^2 + 1)(x + 2)]
    These forms are what one would do, if they had been instructed to "factor the polynomial" or "completely factor the given expression", et cetera.

    Two comments:

    We can see by inspection that this factorization is not correct because we know that foiling means the terms 2x^2 and x will be multiplied together, at some point. That product is 2x^3, which cannot be a term in a quadratic polynomial (aka: 2nd-degree polynomial). In other words, the largest exponent in any quadratic polynomial is always 2.

    Each of these factorizations is the same. Placing an extra set of grouping symbols (the square brackets) around the two factors in parentheses does nothing. It's like the expression 2+2 is exactly the same as (2+2) and [(2+2)]. It probably took me a year in beginning algebra, before I had enough experience to parse algebraic expressions involving multiple grouping symbols, to recognize unnecessary grouping symbols, and to develop my own style of expression using grouping symbols. (Opinions do vary.)

    By the way, it's easier to read polynomials, if you put spaces around the operators (as I added, in both quotes above). It's also not necessary to type the variables in italics. Cheers.
    Last edited by mmm4444bot; 01-26-2018 at 06:02 AM. Reason: inserted spaces in both quotes
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

  3. #3
    Elite Member mmm4444bot's Avatar
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    Factor: 2x^2 + 5x + 3

    I don't know how much you've learned about methods for factoring quadratic polynomials by hand, but the method that I like when A is an Integer not equal to 1 or -1 is called "factor by grouping".

    You can read about it at this site, along with other methods for other types of polynomials. I'm sure Khan Academy also has a video of the factor-by-grouping method.
    "English is the most ambiguous language in the world." ~ Yours Truly, 1969

  4. #4
    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by Quick View Post
    I have no idea what I am getting wrong here...

    here is the problem:

    3) Factor out the greatest common factor in the following polynomial.

    10x​2​​+25x+15
    To learn about finding the greatest common factor in polynomial expressions, please try here. To learn various methods for factoring quadratics, please try here.

    Quote Originally Posted by Quick View Post
    I tried to do two things:

    5(2x^2+1)(x+2) and
    5[(2x^2+1)(x+2)]

    both are wrong, why?
    First, they're the same thing; the only difference is that the second statement has an extra set of grouping symbols. Second, the factoring forward (that is, the "taking out front") of the Greatest Common Factor was done when the 5 was pulled out. (The first link above is a list of lessons that explain the terminology and techniques for finding and factoring out the GCF.)

    Also, the "factorization" of the quadratic is incorrect. This can be shown by multiplying your factors out:

    Code:
    multiplication:
    
           2x^2 + 0x + 1
                  1x + 2
    --------------------
           4x^2 + 0x + 2
    2x^3 + 0x^x + 1x
    --------------------
    2x^3 + 4x^2 + 1x + 2
    Clearly, this is not what resulted when the 5 was pulled out.

    (The second link above provides a list of lessons that explain the terms and techniques for factoring quadratics. To learn how to multiply polynomials, as I did above, please try here.)

  5. #5
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    Ah, I see my mistakes. Thanks for the help.

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