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Math Help - Factor Theorem, help?

  1. #1
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    Factor Theorem, help?

    In an explanation text about “The Factor Theorem” the following example is given:

    Example: A polynomial is given by f(x) = 2x3 + 13x2 + 13x – 10.
    c) Factorise f(x) completely.
    Solution:
    2x3 + 13x2 +13x – 10 = (x + 2)(ax2 +bx + c).
    This is an identity and so the values of a, b and c may be found by comparing coefficients.

    Comparing the x3 term gives: a = 2.
    Comparing the x2 term gives: 13 = 2a + b → b = 9.
    Comparing the constant term gives: c = -5.
    If follows that: f(x) = (x + 2)(2x2 + 9x – 5).
    Factorising the quadratic part in the usual way gives:
    f(x) = (x + 2)(2x – 1)(x + 5).
    My questions are:

    a) where does the expression “2a + b” come from in the line “comparing the x2 term gives” and
    b) how is the c = -5 calculated?

    I can see that 2x3 + 13x2 becomes 2a + b if a is = x3 and b = x2 but why? What are the rules being exercised?

    NB 2x3 is 2x cubed, 13x2 is 13x squared, see attached file
    Many thanks

    Steve Gould
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by stephen_gould View Post
    In an explanation text about “The Factor Theorem” the following example is given:

    Example: A polynomial is given by f(x) = 2x3 + 13x2 + 13x – 10.
    c) Factorise f(x) completely.
    Solution:
    2x3 + 13x2 +13x – 10 = (x + 2)(ax2 +bx + c).
    This is an identity and so the values of a, b and c may be found by comparing coefficients.

    Comparing the x3 term gives: a = 2.
    Comparing the x2 term gives: 13 = 2a + b → b = 9.
    Comparing the constant term gives: c = -5.
    If follows that: f(x) = (x + 2)(2x2 + 9x – 5).
    Factorising the quadratic part in the usual way gives:
    f(x) = (x + 2)(2x – 1)(x + 5).
    My questions are:

    a) where does the expression “2a + b” come from in the line “comparing the x2 term gives” and
    b) how is the c = -5 calculated?
    You have:

    2x^3 + 13x^2 +13x – 10 = (x + 2)(ax^2 +bx + c).

    expanding the right hand side:

    2x^3 + 13x^2 +13x – 10 =ax^3+(2a+b)x^2+(2b+c)x+2c

    Now look at the coefficient of x^2 on both sides of this identity these
    must be equal so:

    13=2a+b.

    And now look at the constant terms on both sides, these also must be
    equal so:

    -10=2c,

    so c=-5.


    RonL
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  3. #3
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    got it, thanks

    Got it thanks.
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