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Math Help - Polynomial functions. Thanks :)

  1. #1
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    Polynomial functions. Thanks :)

    1. The function f(x) = 2x^3 +ax^2 -bx +3 has a factor (x+3). When f(x) is divided by (x-2), the remainder is 15.
    a. Calculate the values of a and b
    b. Find the other two factors of f(x)

    2. Let P(x)=x^5-3x^4+2x^3-2x^2+3x+1
    Given that P(x) can be written in the form (x^2-1)Q(x)+ax +b where Q(x) is a polynomial and a and b are constants, hence or otherwise, find the remainder when P(x) is divided by x^2-1

    3. The expressions px^4-5x+q and x^4-2x^3-px^2-qx-8 have a common factor x-2. Find the values of p and q

    4. Write down one quadratic factor of x^4+x^3-x^2-3x-6, and find a second quadratic factor.

    ps theses are questions from my textbook so i know wot the answers r i just can't get to them. if u don't might could u explain what ur doing.thanks.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by chaneliman View Post
    1. The function f(x) = 2x^3 +ax^2 -bx +3 has a factor (x+3). When f(x) is divided by (x-2), the remainder is 15.
    a. Calculate the values of a and b
    since (x + 3) is a factor, it means that f(-3) = 0 by the factor theorem

    since when f(x) is divided by (x - 2) the remainder is 15, it means f(2) = 15 by the remainder theorem

    this gives you two simultaneous equations from which you can solve for a and b

    b. Find the other two factors of f(x)
    once you find a and b, divide f(x) by (x + 3) using long or synthetic division. the quotient will be a quadratic from which you can find the other two factors. or you can try your luck with the rational roots theorem, but you've probably never seen that
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  3. #3
    Bar0n janvdl's Avatar
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    Quote Originally Posted by chaneliman View Post
    1. The function f(x) = 2x^3 +ax^2 -bx +3 has a factor (x+3). When f(x) is divided by (x-2), the remainder is 15.
    a. Calculate the values of a and b
    b. Find the other two factors of f(x)

    2. Let P(x)=x^5-3x^4+2x^3-2x^2+3x+1
    Given that P(x) can be written in the form (x^2-1)Q(x)+ax +b where Q(x) is a polynomial and a and b are constants, hence or otherwise, find the remainder when P(x) is divided by x^2-1

    3. The expressions px^4-5x+q and x^4-2x^3-px^2-qx-8 have a common factor x-2. Find the values of p and q

    4. Write down one quadratic factor of x^4+x^3-x^2-3x-6, and find a second quadratic factor.

    ps theses are questions from my textbook so i know wot the answers r i just can't get to them. if u don't might could u explain what ur doing.thanks.
    1. This one isn't hard. When x = -3 then f(x) = 0 and when x = 2 then f(x) = 15. Solve the equations simultaneously.

    2. x = \pm 1. Set those values into f(x)

    3. When you set x = 2 then the expressions will both be equal to 0. This is another problem where you should use simultaneous equations.
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by chaneliman View Post
    4. Write down one quadratic factor of x^4+x^3-x^2-3x-6, and find a second quadratic factor.
    The rational root theorem fails to find a rational root for the quartic, so we are essentially left with some sort of "guess and check." However if we are to write a quadratic factor then we know that the polynomial factors as
    x^4+x^3-x^2-3x-6 = (x^2 + ax + b)(x^2 + dx + d)
    where a, b, c, and d are integers.

    So let's expand out the right hand side and compare coefficients with the quartic. This gives the system of equations:
    a + c = 1

    ac + b + d = -1

    ad + bc = -3

    bd = -6

    I think the easiest way to solve this is to solve the top and bottom equations for c and d respectively, giving
    -a^2 + a +b - \frac{6}{b} = -1

    -a \left ( b + \frac{6}{b} \right ) + b = -3

    Now, if we are going to have integer values for b, then from bd = -6 we know that b can only b \pm 1, \pm 2, \pm 3, \pm 6.

    From this I get solutions
    a = 1, b = 2
    a = 0, b = -3

    So
    x^4+x^3-x^2-3x-6 = (x^2 + x + 2)(x^2 - 3)

    -Dan
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