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Math Help - Simplification!

  1. #1
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    Post Simplification!

    I was wondering If i would be able to get help with these last questions:

    a) Prove by showing all steps of working that 3^(x+2)+27 / 5 x 3^x + 15 can be simplified to 9 / 5

    a) p ( x) = ax^3 + bx^2 - 5x + c is a polynomial exactly divisible by (x+2).
    Also p(0) = p (1) = -6, find a, b and c and completely factorize p (x).

    Any Contribution will be appreciated!
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  2. #2
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    3^(x+2) + 27 = 3^2(3^x + 3) = 9(3^x + 3)

    5(3^x) + 15 = 5(3^x + 3)

    Help?
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  3. #3
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    Hello, Pandp77!

    p(x) \:= \:ax^3 + bx^2 - 5x  + c is a polynomial exactly divisible by (x+2).

    Also: . p(0) \:= \:p(1) \:= \:-6

    Find a, b\text{ and }c, and completely factorize p(x)

    Since p(x) is divisible by (x+2), then: . f(\text{-}2) \:=\:0
    . . a(\text{-}2)^3 + b(\text{-}2)^2 - 5(\text{-}2) + c \:=\:0 \quad\Rightarrow\quad \text{-}8a + 4b + c \:=\:\text{-}10\;\;{\color{blue}[1]}

    Since p(0) = \text{-}6\!:\;\;a(0^3) + b(0^2) - 5(0) + c \:=\:\text{-}6\quad\Rightarrow\quad\boxed{ c \:=\:-6}\;\;{\color{blue}[2]}

    Since p(1) = \text{-}6\!:\;\;a(1^3) + b(1^2) - 5(1) + c \:=\:\text{-}6\quad\Rightarrow\quad a + b + c \:=\:\text{-}1\;\;{\color{blue}[3]}


    Substitute [2] into [1]: . \text{-}8a + 4b -6 \:=\:\text{-}10 \quad\Rightarrow\quad 2a - b \:=\:1\;\;{\color{blue}[4]}

    Substitute [2] into [3]: . . a + b -6 \:=\:-1\quad\Rightarrow\quad\;\; a + b \:=\:5\;\;{\color{blue}[5]}

    Add [4] and [5]: . 3a \:=\:6 \quad\Rightarrow\quad \boxed{a \:=\:2}

    Substitute into [5]: . 2 + b \:=\:5\quad\Rightarrow\quad\boxed{ b \:=\:3}


    Hence: . p(x) \;=\;2x^3 + 3x^2 - 5x - 6

    We know that p(x) is divisible by (x+2)\!: \;\;p(x) \;=\;(x+2)(2x^2 - x - 3)


    Therefore: . p(x) \;=\;(x+2)(x+1)(2x-3)

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