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    Practice Problems - Polynomials and Rational Functions

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    1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished.

    Use synthetic division to find the function value.
    1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2).

    Write the quadratic function in the form y = a(x - h)2 + k.
    2) y = x2 - 2x - 9

    Find the product.
    3) [x - (-6 + √13)][x - (-6 - √13)]

    Use the leading coefficient test to determine whether y→∞ or y→-∞ as x→ -∞.
    4) y = -5x3 + 4x2 + 6x - 7

    For the given function, find all asymptotes of the type indicated (if there are any).
    5) f(x) = (x-9)/(x^2- 4) vertical

    Use the rational zero theorem to find all possible rational zeros for the polynomial function.
    6) P(x) = 3x3 + 43x2 + 43x + 27

    Solve the inequality. Give answer in interval notation.
    7) (x + 2)(x - 1)(x - 10) > 0

    Solve the inequality.
    8) (x+21)/(x+3) <2

    Discuss the symmetry of the graph of the polynomial function.
    9) f(x) = x2 + 2x - 1

    Solve the absolute value equation.
    10) |x2 - 10| = 4

    Solve the quadratic inequality by graphing an appropriate quadratic function.
    11) x2 - 2x - 8 ≤ 0

    Use the theorem on bounds to establish the best integral bounds for the roots of the equation.
    12) 6x3 - 7x2 + 7x + 9 = 0

    State the degree of the polynomial equation.
    13) 4(x + 8)2(x - 8)3 = 0

    © BrainMass Inc. brainmass.com October 5, 2022, 5:34 pm ad1c9bdddf
    https://brainmass.com/math/number-theory/practice-problems-polynomials-rational-functions-174075

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    1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished.

    Let us use synthetic division to determine if the given set of numbers are zeros of the given polynomial
    {-3, 2} f(x) = 3x^3 + 5x^2 - 6x + 18.

    3 3 5 -6 18
    -9 12 -18
    3 4 6 0
    Since the last number (remainder) is 0, x = -3 is a zero of f(x) = 0  x + 3 is a factor of f(x)
    2 3 5 -6 18
    6 22 32
    3 11 16 50
    Since the last number (remainder) is nonzero, x = 2 is not a zero of f(x) = 0  x - 2 is a not factor of f(x) ANSWERS:
    Synthetic division has been illustrated.

    Use synthetic division to find the function value.
    1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2).

    -2 2 4 2 3 8
     -4 0 -4 2
    2 0 2 -1 10
    ANSWERS:
    f(-2) = Remainder = 10

    Write the quadratic function in the form y = a(x - h)2 + k.
    2) y = x2 - 2x - 9

    y = x^2 - 2x - 9
    y = x^2 - 2x + 4 - 13
    y = (x - 2)^2 - 13
    This is of the form y = a(x - h)^2 + k, where
    a = 1, h = 2, k = -13 ANSWERS:
    y = (x - 2)^2 - 13

    Find the product.
    3) [x - (-6 + √13)][x - (-6 - √13)]

    Given expression = (x + 6 - Ö13)(x + 6 + Ö13)
    = (x + 6)^2 - (Ö13)^2
    = x^2 + 12x + 36 - 13
    = x^2 + 12x + 23 ANSWERS:

    Use the leading coefficient test to determine whether y→∞ or y→-∞ as x→ -∞.
    4) y = -5x3 + 4x2 + 6x - 7

    Since the leading coefficient is negative, as x  -∞,
    y  ∞ ANSWERS:
    y  ∞

    For the given function, find all asymptotes of the type indicated (if there are any).
    5) f(x) = (x-9)/(x^2- 4) vertical

    f(x) = (x - 9)/(x + 2)(x - 2)
    f(x) has vertical tangents at x = {-2, 2} ANSWERS:
    x = -2 and x = 2

    Use the rational zero theorem to find all possible rational zeros for the polynomial function.
    6) P(x) = 3x3 + 43x2 + 43x + 27

    The possible rational zeros of P(x) are given by
    ± (factors of 27)/factors of 3
     ± (1, 3, 9, 27)/(1, 3)
     {± 1, ± 3, ± 9, ± 27, ± 1/3} ANSWERS:
    {± 1, ± 3, ± 9, ± 27, ± 1/3}

    Solve the inequality. Give answer in interval notation.
    7) (x + 2)(x - 1)(x - 10) > 0

    Given expression = (x + 2)(x - 1)(x - 10)
    The critical numbers are x = {-2, 1, 10}
    For x < -2, (x + 2)(x - 1)(x - 10) < 0
    For -2 < x < 1, (x + 2)(x - 1)(x - 10) > 0
    For 1 < x < 10, (x + 2)(x - 1)(x - 10) < 0
    For x > 10, (x + 2)(x - 1)(x - 10) > 0
     The solution is (-2, 1) È (10, ∞) ANSWERS:
    (-2, 1) È (10, ∞)

    Solve the inequality.
    8) (x+21)/(x+3) <2

    (x + 21/(x + 3) < 2
    (x + 21)(x + 3) < 2(x + 3)^2
    0 < (x + 3)[2(x + 3) - (x + 21)]
    0 < (x + 3)(x - 15)
    (x + 3)(x - 15) > 0
     x < -3 or x > 15 ANSWERS:
    (-∞, -3) È (15, ∞)

    Discuss the symmetry of the graph of the polynomial function.
    9) f(x) = x2 + 2x - 1

    f(-x) = x^2 - 2x - 1 and -f(x) = -x^2 - 2x + 1
    = -(x^2 + 2x - 1)
    We see that f(-x) ≠ f(x) and also f(-x) ≠ -f(x) ANSWERS:
    f(x) has no symmetry of any kind.

    Solve the absolute value equation.
    10) |x2 - 10| = 4

    |x^2 - 10| = 4  x^2 - 10 = ± 4
    x^2 = 10 ± 4
    x^2 = 14 or 6
    x = Ö14 or x = Ö6 ANSWERS:
    x = {± Ö6, ± Ö14}

    Solve the quadratic inequality by graphing an appropriate quadratic function.
    11) x2 - 2x - 8 ≤ 0

    ANSWERS:
    The solution is [-2, 4]

    Use the theorem on bounds to establish the best integral bounds for the roots of the equation.
    12) 6x3 - 7x2 + 7x + 9 = 0

    The possible rational factors of
    f(x) = 6x^3 - 7x^2 + 7x + 9 are given by
    {± (factors of 9)/factors of 6}
    = {± (1, 3, 9)/(1, 2, 3, 6)}
    = {± 1, ± 3, ± 9, ± ½, ± 3/2, ± 9/2, ± 1/3, ± 1/6}
    There are 2 sign changes in f(x)  f(x) has 0 positive roots or 2 positive roots. In fact f(x) has two complex roots.
    f(-x) = -6x^3 - 7x^2 - 7x + 9
    There is just one sign change in f(-x)  f(x) has one negative root
    From the list of possible roots, choose x = 1/6 and
    x = -1
    -1 6 -7 7 9
     -6 13 -20
    6 -13 20 -11
    Since the entries in the last row alternate in sign,
    x = -1 is a lower bound for the roots of f(x)
    -1/2 6 -7 7 9
     -3 5 -6
    6 -10 12 3
    x = -1/2 is an upper bound ANSWERS:
    x = -1 is a lower bound and x = -1/2 is an upper bound.

    State the degree of the polynomial equation.
    13) 4(x + 8)2(x - 8)3 = 0
    4(x + 8)^2 (x - 8)^3 = 0 has a degree 2 + 3 = 5 ANSWERS:
    The degree is 5.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com October 5, 2022, 5:34 pm ad1c9bdddf>
    https://brainmass.com/math/number-theory/practice-problems-polynomials-rational-functions-174075

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