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Math Help - Polynomials questions again... part 5

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    Polynomials questions again... part 5

    I don't know how to solve this! Can someone show the formula?

    1. Find all possibles values of k such that the equation x^2-(k-3)x+k^2+2k+5=0 has real roots. If \alpha and \beta are real roots of this equation, show that \alpha^2+\beta^2=-(k+5)^2+24. Hence, find the maximum value \alpha^2+\beta^2.

    2a. If p(x)=3x^2+5x-2 is a factor of the polynomial q(x)=3x^4+8x^3+kx^2+3x-2, determine the value of k. With this value of k, find all four zeroes of q(x).

    2b. Find the solution set of the inequality x[p(x)]>0.

    Answer:
    1. \{k:-\frac{11}{3}\leq k \leq -1\}; Maximum value = \frac {200}{9}

    2a. k=6; -2, \frac{1}{3}, -\frac{1}{2}\pm \frac{\sqrt{3}}{2}i

    2b. \{x:-2<x<0 or x>\frac{1}{3}\}
    Last edited by cloud5; July 15th 2009 at 05:39 AM.
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    Quote Originally Posted by cloud5 View Post
    I don't know how to solve this!

    1. Find all possibles values of k such that the equation x^2-(k-3)x+k^2+2k+5=0 has real roots. If \alpha and \beta are real roots of this equation, show that \alpha^2+\beta^2=-(k+5)^2+24. Hence, find the maximum value \alpha^2+\beta^2.

    2a. If p(x)=3x^2+5x-2 is a factor of the polynomial q(x)=3x^4+8x^3+kx^2+3x-2, determine the value of k. With this value of k, find all four zeroes of q(x).

    2b. Find the solution set of the inequality x[p(x)]>0.

    Answer:
    1. \{k:-\frac{11}{3}\leq k \leq -1\}; Maximum value = \frac {200}{9}

    2a. k=6; -2, \frac{1}{3}, -\frac{1}{2}\pm \frac{\sqrt{3}}{2}i

    2b. \{x:-2<x<0 or x>\frac{1}{3}\}
    For number 1 note that it's a quadratic in x so use the discriminant (b^2-4ac) and set it greater than 0 to solve for real roots
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  3. #3
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    Quote Originally Posted by cloud5 View Post
    I don't know how to solve this!

    1. Find all possibles values of k such that the equation x^2-(k-3)x+k^2+2k+5=0 has real roots. If \alpha and \beta are real roots of this equation, show that \alpha^2+\beta^2=-(k+5)^2+24. Hence, find the maximum value \alpha^2+\beta^2.

    2a. If p(x)=3x^2+5x-2 is a factor of the polynomial q(x)=3x^4+8x^3+kx^2+3x-2, determine the value of k. With this value of k, find all four zeroes of q(x).

    2b. Find the solution set of the inequality x[p(x)]>0.

    Answer:
    1. \{k:-\frac{11}{3}\leq k \leq -1\}; Maximum value = \frac {200}{9}

    2a. k=6; -2, \frac{1}{3}, -\frac{1}{2}\pm \frac{\sqrt{3}}{2}i

    2b. \{x:-2<x<0 or x>\frac{1}{3}\}
    Q1. Start by using the fact that you require the discriminant of the quadratic to be greater than zero.

    Q2 a. Start by noting that 3x^2 + 5x - 2 = (3x - 1)(x + 2). Therefore q(-2) = 0.

    Q2b. Start by taking a graphical approach and sketch the graph of y = x p(x). For what values of x is y > 0?
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