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Math Help - Factorisation

  1. #1
    Junior Member Pinsky's Avatar
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    Factorisation

    I have s^3+12s^2+21s+10 and a have to show it as factors.
    The resut should be (s+1)^2(s+10).

    What is the procedure?
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  2. #2
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    Hello, Pinsky!

    Factor: . f(s) \:=\:s^3+12s^2+21s+10

    Answer: . (s+1)^2(s+10).

    What is the procedure?
    We are expected to be familiar with two theorems . . .


    [1] The rational roots of a polynomial are of the form \frac{n}{d}
    . . .where n is a factor of the constant term and d is a factor of the leading coefficient.

    Our polynomial has a constant term 10 with factors: . \pm1,\:\pm2,\:\pm5,\:\pm10
    . . and leading coefficient 1 with factors: . \pm1
    Hence, the only possible rational roots are: . \pm1,\:\pm2,\:\pm5,\:\pm10


    [2] If f(a) = 0, then (x-a) is a factor of f(x).

    We find that: . f(\text{-}1) \:=\:(\text{-}1)^3 + 12(\text{-}1)^2 + 21(\text{-}1) + 10 \:=\:0
    . . Hence, (s+1) is a factor.

    Using long or synthetic division: . f(s) \:=\:(s+1)(s^2+11s + 10)

    . . And that quadratic factors: . s^2+11s + 10 \:=\:\overbrace{(s+1)(s+10)}


    Therefore: . f(s) \;=\;(s+1)^2(s+10)

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