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Math Help - Please help me with this sample test

  1. #1
    Junior Member
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    Please help me with this sample test

    Hello everybody, could you please give me a hand with this sample test

    1-form a polynomial whose zeros and degree are given

    zeros : -1, 1, 3; degree 3

    2-for the polynomial function, a- list each zero and its multiplicity, b= determine whether the graph croses or touches the x axis at each x intercept. C find the power function that the graph of f resembles for large values of |x|

    f(x)= (x-5)^3 (x+4)^2

    3- for the given polynomial function f:

    a-find the x and y intercepts of f
    b-determine whether the graph of f crosses or touches the x axis at each x intercept
    c-end behavior. Find the poer funcion that the graph of f resembles for large values of |x|
    d- determine the maximum number of turning points on the graph of f
    e- Use the x intercepts to find the intervals on which the graph of f is above and below the x axis
    f- plot the points obtained in parts a- and e- and use the remaining information to conect them with a smooth, continuos curve

    F(x)= (x-1)(x-2)(x-4)


    4- find the vertical, horizontal and oblique asymptopes, if any, in the given rational function

    3x+5

    5-find the vertical, horizontal and oblique asymptotes, of the given rational function

    f(x)-=3x+5/x-6

    6-Use the factor theorem to determine whether x-c is a factor of f(x)

    f(x) = -4x^7+x^3-x^2+2


    7-List the potential zeros of each polynomial function
    f(x)=6x^4-x^2+9

    8-Use Descartes' rule of signs and the rational zeros theorem to find all real zeros in the given polynomial functions, use the zeros to factor f over the real numbers

    f(x)= X^3+2x^2-5x-6
    f(x)=2x^3+x^2+2x+1

    9-Solve the given equation in the real number system

    2x^3-3x^2-3x-5=0

    10-
    Use the intermediate value theorem to show that given polynomial function has a zero in the given interval

    f(x) = 8x^4-2x^2+5x-1 {0,1}

    11-In the following problem information is given about the polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f

    Degree 4; zeros: i,1+i


    12-Form a polynomial f(x) with real coefficients having the given degree and zeros

    Degree 5; zeros: 2, -i;1+i

    13-Use the given zero to find the remaining zeros of the given function

    F(x) = 2x^4+5x^3+5x^2+20x-12 ; Zero: -2i
    14-
    Find the complex zeros of the given polynomial function. Write f in factored form

    F(x) = x^4+5x^2+4


    Thank you so much.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jhonwashington View Post
    1-form a polynomial whose zeros and degree are given

    zeros : -1, 1, 3; degree 3
    This will be a cubic, so (x-(-1))(x-1)(x-3) is one such cubic, just multiply
    out the brackets.

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by jhonwashington View Post
    13-Use the given zero to find the remaining zeros of the given function

    F(x) = 2x^4+5x^3+5x^2+20x-12 ; Zero: -2i
    As the polynomial has real coefficients the complex roots exist as conjugate
    pairs, so if x=-2i is a root so is x=2i, which means that (x-2i)(x+2i)=x^2+4 is
    a factor of F(x).

    So dividing F(x) by x^2+4, we find:

    2x^4+5x^3+5x^2+20x-12 = (x^2+4)(2x^2+5x -3)

    Now use the quadratic formula to find the roots of 2x^2+5x -3, which are
    x=1/2 and x=-3.

    Thus the roots of F(x) are -2i, 2i, 1/2 and -3 (which we know is a complete list
    of the roots of F(x), as every polynomial of degree n has at most n complex roots).

    RonL
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  4. #4
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    Hello, jhonwashington!

    7) List the potential zeros of this polynomial function: .f(x) .= .6x^4 - x^2 + 9
    A potential zero has the form: n/d
    . . where n is a factor of the constant term
    . . and d is a factor of the leading coefficient.

    The factors of 9 are: 1, 3, 9
    The factors of 6 are: 1, 2, 3, 6

    The potential zeros are: 1, 3, 9, 1/2, 3/2, 9/2, 1/3, 1/6



    9) Solve the given equation in the real number system: .2x - 3x - 3x - 5 .= .0
    Using the Remainder Theorem, we find that x = 5/2 is a zero.
    . . Hence, (2x - 5) is a factor.

    Then we have: .(2x - 5)(x + x + 1) .= .0

    The quadratic factor has no real roots.

    Therefore, the only real root is: .x = 5/2



    12) Form a polynomial f(x) with real coefficients having the given degree and zeros:
    . . . Degree 5; zeros: 2, -i; 1 + i

    Complex roots always appear in conjugate pairs.
    If i is a zero, then -i is a zero.
    If 1 + i is a zero, then 1 - i is a zero.

    The polynomial is: .(x - 2) (x - i) (x + i) (x - [1 + i]) (x - [1 - i])

    I'll let you multiply it out . . .



    14) Find the complex zeros of the given polynomial function. Write F in factored form.
    . . F(x) .= .x^4 + 5x^2 + 4

    We have: .x^4 + 5x^2 + 4 .= .0

    Factor: .(x + 1)(x + 4) .= .0

    Then: .x + 1 .= .0 . . x = -1 . . x = i
    . . . . . x + 4 .= .0 . . x = -4 . . x = 2i

    . . F(x) .= .(x - i)(x + i)(x - 2i)(x + 2i)

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