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Thread: Polynomial Problems

  1. #1
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    Polynomial Problems

    Hello, i have some problem with these two polynomial questions.

    1.

    The polynomial 6x3 + mx2 + nx - 5 has a factor of x+1. When divided by x-1, the remainder is 4. What are the values of m and n?

    I am lost completely on this one..

    2.

    Hyun has designed a rectangular building. His scale model is 1m x 2m x 4m. In order to maintain the scale he needs to increase each dimension by the same amount. By what amount should he increase each dimension to make a building that is no more than 9 times the volume of his scale model? Give a full algebraic solution.

    For this one i have found the amount that needs to be increased (2m for each dimension) and have figured out that it is a quartic function from the 4th differences, as well as the y intercept being 8 but i do not know what to do from here.

    Any help is appreciated thank youu!!!!
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  2. #2
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    Re: Polynomial Problems

    #1 is a cool problem. We are going to construct a system of 2 equations with 2 unknowns, namely m and n. We will do this in two parts.

    PART 1: We are given that P(x)=6x^3+mx^2+mx-5 has a factor of x+1. This means that P(x) has remainder 0 when divided by x+1.

    Perform the synthetic division as shown under the title "Regular synthetic division" at https://en.wikipedia.org/wiki/Synthe...hetic_division.

    We will collect a remainder in terms of m and n, and then set said remainder equal to 0.

    PART 2: We are also given that the quotient \dfrac{P(x)}{x-1} has remainder 4. As such, we perform the synthetic division this time using the divisor x-1. Again, we get a remainder in terms of m and n, which we now set equal to 4.

    PUTTING IT TOGETHER: In Part 1, I got

    \begin{align*}\dfrac{6x^3+mx^2+nx-5}{x+1} = 6x^2+(m-6)x+(n-m+6) + \dfrac{m-n-11}{x+1}\text{.}\end{align*}

    So \dfrac{P(x)}{x+1} has remainder m-n-11. Remember that this remainder will be set equal to 0.

    In Part 2, I got

    \begin{align*}\dfrac{6x^3+mx^2+nx-5}{x-1} = 6x^2+(m+6)x+(m+n+6) + \dfrac{m+n+1}{x-1}\text{.}\end{align*}

    So \dfrac{P(x)}{x-1} has remainder m+n+1. Remember that this remainder will be set equal to 4.

    We now have
    \begin{array}{lcl} m-n-11 & = & 0 \\ m+n+1 & = & 4\text{.} \end{array}

    Solve this system and you are done

    And please double-check my calculations.

    Good luck!
    -Andy
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  3. #3
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    Re: Polynomial Problems

    Quote Originally Posted by SillyBilly View Post
    Hello, i have some problem with these two polynomial questions.

    1.

    The polynomial 6x3 + mx2 + nx - 5 has a factor of x+1. When divided by x-1, the remainder is 4. What are the values of m and n?

    I am lost completely on this one..

    2.

    Hyun has designed a rectangular building. His scale model is 1m x 2m x 4m. In order to maintain the scale he needs to increase each dimension by the same amount. By what amount should he increase each dimension to make a building that is no more than 9 times the volume of his scale model? Give a full algebraic solution.

    For this one i have found the amount that needs to be increased (2m for each dimension) and have figured out that it is a quartic function from the 4th differences, as well as the y intercept being 8 but i do not know what to do from here.

    Any help is appreciated thank youu!!!!
    2. Whatever the length scaling factor is, the volume is scaled by the cube of this factor, or conversely, whatever the volume is scaled by, the length is scaled by the cube root of.
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  4. #4
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    Re: Polynomial Problems

    Quote Originally Posted by SillyBilly View Post
    Hello, i have some problem with these two polynomial questions.
    1.The polynomial 6x3 + mx2 + nx - 5 has a factor of x+1. When divided by x-1, the remainder is 4. What are the values of m and n?
    Lets say $P(x)=6x^3+mx^2+nx-5$
    From the given we know that $P(-1)=0~\&~P(1)=4$
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