2. Polynomial Functions
  3. All Exercises, Answers, and Solutions

Answers


Transformations of Simple Polynomial Functions


    1. Reflected in the \( y \)-axis, stretched horizontally about the \( y \)-axis by a factor of \( \frac{1}{3} \), translated left \( 1 \) unit and down \( 2 \) units
    2. Reflected in the \( x \)-axis, stretched vertically about the \( x \)-axis by a factor of \( \frac{1}{5} \), translated right \( 4 \) units and up \( 1 \) unit.
    1. IV
    2. I
    3. II
    4. III
      1. \( y = x^3 \); reflected in the \( x \)-axis, stretched vertically about the \( x \)-axis by a factor of \( 2 \), translated left \( 3 \) units
      2. \( y = x^4 \); stretched vertically about the \( x \)-axis by a factor of \( \frac{1}{2} \), translated down \( 5 \) units
      3. \( y = x^3 \); stretched horizontally about the \( y \)-axis by a factor of \( 2 \), translated right \( 2 \) units
      4. \( y = x^4 \); reflected in the \( x \)-axis, stretched horizontally about the \( y \)-axis by a factor of \( \frac{1}{2} \), translated up \( 7 \) units
      1. Domain: \( \{ x \mid x \in \mathbb{R} \} \)
        Range: \( \{ y \mid y \in \mathbb{R} \} \)
        Graph of y = x^3 and y = -2(x+3)^3
      2. Domain: \( \{ x \mid x \in \mathbb{R} \} \)
        Range: \( \{ y \mid y \geq -5, y \in \mathbb{R} \} \)
        Graph of y = x^4 and y = -0.5x^4 - 5
      3. Domain: \( \{ x \mid x \in \mathbb{R} \} \)
        Range: \( \{ y \mid y \in \mathbb{R} \} \)
        Graph of y = x^3 and y = (0.5x - 1)^3
      4. Domain: \( \{ x \mid x \in \mathbb{R} \} \)
        Range: \( \{ y \mid y \leq 7, y \in \mathbb{R} \} \)
        Graph of y = x^4 and y = -(2x)^4 + 7
    1. \( (x, y) \rightarrow (x, 3y + 2), y = 3x^3 + 2 \)
    2. \( (x, y) \rightarrow (4x - 2, y), y = \left(\frac{1}{4}(x + 2)\right)^3 \)
    3. \( (x, y) \rightarrow (-x + 3, \frac{1}{2}y), y = \frac{1}{2}\left( -(x - 3) \right)^3 \)
    4. \( (x, y) \rightarrow (\frac{1}{4}x, -y + 2), y = -\left(4x\right)^3 + 2 \)
    1. \( y = -6\left(x - 2\right)^3 + 4 \)
    2. \( y = -3\left(x - 4\right)^4 + 2 \)
    1. \( f(x) = 2x^3 - 5 \)
    2. \( f(x) = -\frac{8}{81}(x - 2)^4 - 3 \)
    3. \( f(x) = (x + 5)^4 + 2 \)
    1. \( y = -3(x + 2)^3 - 3 \)
    2. \( y = -\frac{1}{2}(x - 4)^4 + 5 \)
    1. \( (x, y) \rightarrow ( x + 2, y + 4 ) \)
    2. \( (x, y) \rightarrow ( x, \frac{5}{2}y - 7 ) \)
    3. \( (x, y) \rightarrow ( x - \frac{5}{2}, 2y - 1 ) \)
    4. \( (x, y) \rightarrow ( -\frac{1}{2}x + \frac{5}{2}, -y + 8 ) \)
    1. \( (x + 2, y + 1) \)
    2. \( (x, 5y) \)
    3. \( (2(x - 2), -y) \)
    4. \( \left( x - 3, -2(y - 4) \right) \)
    1. One
    2. Zero
    3. One
    4. Zero
    1. \( y = -5x(x + 6)(x + 1) - 5 \)
    2. Vertical stretch about the \( x \)-axis by factor of \( \frac{1}{6} \); horizontal translation \( 1 \) unit left
  12. \( f(x) = x^3 + 10, g(x) = -\frac{2}{27}(x - 5)^3 - 21 \)

Characteristics of Polynomial Functions


      1. End behaviour: As \( x \rightarrow \infty \), \( y \rightarrow -\infty \); as \( x \rightarrow -\infty \), \( y \rightarrow \infty \)
      2. Maximum \( 2 \) turning points, minimum \( 0 \) turning points
      3. Maximum \( 3 \) zeros, minimum \( 1 \) zero
      1. End behaviour: \( y \rightarrow \infty \) as \( x \rightarrow \pm \infty \)
      2. Maximum \( 3 \) turning points, minimum \( 1 \) turning point
      3. Maximum \( 4 \) zeros, minimum \( 0 \) zeros
      1. End behaviour: As \( x \rightarrow \infty \), \( y \rightarrow \infty \); as \( x \rightarrow -\infty \), \( y \rightarrow -\infty \)
      2. Maximum \( 4 \) turning points, minimum \( 0 \) turning points
      3. Maximum \( 5 \) zeros, minimum \( 1 \) zero
    1. Minimum degree of four, leading coefficient negative
    2. Minimum degree of three, leading coefficient positive
    3. Minimum degree of six, leading coefficient positive
    4. Minimum degree of five, leading coefficient negative
    1. False
    2. True
    3. True
    4. True
    5. False
    6. False
    1. IV
    2. II
    3. VI
    4. III
    5. I
    6. V
  5. \( 0, 1, 2, 3, \) or \( 4 \) distinct zeros.
    No zeros: Quartic, negative leading coefficient, no zeros
    One zero: Quartic, negative leading coefficient, one zero
    Two zeros: Quartic, negative leading coefficient, two zeros
    Three zeros: Quartic, negative leading coefficient, three zeros
    Four zeros: Quartic, negative leading coefficient, four zeros
  6. Answers may vary.
    1. Degree 3, zeros at x=-2,1, 2 turning points, opposite end behaviour
    2. Degree 4, zeros at x=-2,2, 3 turning points, posiitve leading coefficient
    3. Degree 4, negative leading coefficient, zeros at x=-2,0,2, 3 turning points
    4. Degree 3, positive leading coefficient, one zero, two turning points
    5. Degree 5, -ve leading coefficient, order 2 zero at x=-2, order 3 zero at x=1, 2 turning pts
    6. Degree 5, +ve leading coefficient, one zero, four turning points
  7. Answers may vary.
    1. Quartic, zeros at x=-4,-1,1,3, positive leading coefficient
      End behaviour: \( y \rightarrow \infty \) as \( x \rightarrow \pm \infty \)
      \( x \)-intercepts: \( -4, -1, 1, 3 \)
      \( y \)-intercept: \( 12 \)
    2. Cubic, zeros at x=-3,0,1.5, negative leading coefficient
      \( y = -x(2x - 3)(x + 3)\)
      End behaviour: \( y \rightarrow -\infty \) as \( x \rightarrow \infty \), \( y \rightarrow \infty \) as \( x \rightarrow -\infty \)

      \( x \)-intercepts: \( -3, 0, \frac{3}{2} \)
      \( y \)-intercept: \( 0 \)
  8. Answers may vary.
    1. Graph of y = |x^3 + x^2 - 6x|
    2. Graph of y = |x|^3 + |x|^2 - 6|x|

Graphs of Polynomial Functions in Factored Form


    1. V
    2. VI
    3. III
    4. I
    5. VII
    1. Answers may vary.
      Quadratic with turning point tangent to x-axis at x = -5, touching from above.
    2. Answers may vary.
      Positive quintic with turning point tangent to x-axis at x = 0, touching from below; zero parallel to and passing through x-axis at x = 3.
    3. Answers may vary.
      Negative cubic with single zero at x = 4, two turning points.
    4. Answers may vary.
      Negative cubic with single zero at x = 4, two turning points.
    5. Answers may vary.
      Positive quintic with turning point tangent to x-axis at x = 1, touching from above, and passing through x-axis at x = -2
    1. \( f(x): \) degree \( 3 \)
      \( g(x): \) degree \( 5 \)
    2. \( f(x): \) \( x \)-intercepts at \( -5, 0, \frac{2}{3} \), \( y \)-intercept at \( 0 \)
      \( g(x): \) \( x \)-intercepts at \( 0, 2 \), \( y \)-intercept at \( 0 \)
    3. \( f(x): \) \( x \rightarrow -\infty, y \rightarrow \infty; x \rightarrow \infty, y \rightarrow -\infty \)
      \( g(x): \) \( x \rightarrow -\infty, y \rightarrow -\infty; x \rightarrow \infty, y \rightarrow \infty \)
    4. \( f(x) \): Graph of f(x) = -2x(3x - 2)(x + 5)
      \( g(x) \): Graph of g(x) = 3x^2(x - 2)^3
    1. Answers may vary.
      Sketch of y = -x(x + 2)(2x - 5)
    2. Answers may vary.
      Sketch of y = 2(x - 2)^2(x + 3)^2
    3. Answers may vary.
      Sketch of y = 0.5(3 - x)(x + 1)^3
    4. Answers may vary.
      Sketch of y = 2x^2(x - 4)^3
    5. Answers may vary.
      Sketch of y = -x(2x + 3)(x - 2)^2
    1. Answers vary. \( y = x(x - 1)^2, y = x^2(x - 1) \)
    2. Answers vary. \( y = (x + 1)^2(x - 1)^2, y = x(x - 1)^3 \)
    3. Answers vary. \( y = -(x - 1)^3(x + 1)^2, y = (x - 1)(x + 1)^4 \)
  6. Answers may vary.
    1. Sketch of f(x) = -2x^3 + 8x
    2. Sketch of f(x) = -x^4 - 5x^3 - 6x^2
    3. Sketch of f(x) = x^4 - 2x^2 + 1
    1. \( x \)-intercepts at \( -4, 0, 2 \); changes sign at \( x = 0, 2 \)
    2. \( x \)-intercepts at \( -3, 4 \); changes sign at \( x = -3 \)
    3. \( x \)-intercepts at \( -4, 1 \); does not change sign; \( y \leq 0 \) for all \( x \in \mathbb{R} \)
    1. Positive on \( x \in (-4, -1) \cup (2, \infty) \); negative on \( x \in (-\infty, -4) \cup (-1, 2) \)
    2. Positive on \( x \in (0, 4) \); negative on \( x \in (-\infty, -3) \cup (-3, 0) \cup (4, \infty) \)
    3. Positive on \( x \in (-\infty, 0) \cup (0, 3) \); negative on \( x \in (3, \infty) \)
    1. From the graph,
      Graph of y = (1/2)(x + 3)(x - 3)^2
      Increasing for \( x \in (-\infty, -1) \cup (3, \infty) \); decreasing for \( x \in (-1, 3) \)
      1. \( -16 < c < 0 \)
      2. \( c = 0 \) or \( c = -16 \)
      3. \( c < -16 \) or \( c > 0 \)
      4. no values for \( c \)
    1. \( f(x) = k(x + 3)(x - 1)^2(x - 4)^2 \)
    2. Two such functions are (note that \( k < 0 \) ):
      \( f(x) = -(x + 3)(x - 1)^2(x - 4)^2 \)
      \( g(x) = -2(x + 3)(x - 1)^2(x - 4)^2 \)
  11. Answers may vary, but are of the form \( y = kx^3(x + 3)(x - 4)^2 \) for \( k < 0, k \in \mathbb{R} \).

Equations of Polynomial Functions in Factored Form


    1. V
    2. II
    3. VIII
    4. VI
    5. III
    6. IV
    7. I
    8. VII
    1. \( y = k(x + 3)(2x + 1)(3x - 5), k \neq 0, k \in \mathbb{R} \)
    2. \( y = k(x - 1)(x + 2)^2(x - 5)^3, k \neq 0, k \in \mathbb{R} \)
    3. \( y = kx(3x - 2)^3, k \neq 0, k \in \mathbb{R} \)
    4. \( y = k(x + 4)(x - 1)^2, k \lt 0, k \in \mathbb{R} \)
    5. \( y = k(x^2 - 5)(x^2 + 2x - 1), k \neq 0, k \in \mathbb{R} \)
    1. \( y = kx(x + 6)(x - 4), k \gt 0, k \in \mathbb{R} \)
    2. \( y = k(x + 7)^2(x - 6), k \lt 0, k \in \mathbb{R} \)
    3. \( y = kx^3(x - 4), k \lt 0, k \in \mathbb{R} \)
    4. \( y = k(x + 3)^2(x - 4)^2, k \lt 0, k \in \mathbb{R} \)
    5. \( y = k(x + 3)(x - 3)^3, k \gt 0, k \in \mathbb{R} \)
    6. \( y = k(x + 4)(x + 2)(x+1)(x - 4)(x - 3)^2, k \lt 0, k \in \mathbb{R} \)
  4. \( y = -\frac{1}{5}(2x + 1)(x - 5)(x - 2)^2 \)
  5. \( y = \frac{1}{2}(x + 1)(x - 4)(x - 6) \)
    1. \( y = k(x^2 + 6x + 4) \)
    2. \( y = kx(x^2 - 2x - 11) \)
    3. \( y = k(x + 2)(x - 1)(x^2 + 9) \)
    4. \( y = k(x^2 - 6x + 7)(x^2 + 8x + 19) \)
  7. Using zeros \( 4 \pm \sqrt{2}, -3 \pm \sqrt{6} \) and \( y \)-intercept \( (0, -21) \), \( y = -\frac{1}{2} (x^2 - 8x + 14)(x^2 + 6x + 3) \)
    1. Positive: \( \{ x \mid x \neq -2, 2, x \in \mathbb{R} \} \)
      Negative: No values of \( x \) for which \( f(x) \) is negative.
    2. Increasing: \( \{ x \mid x \in (-2, 0) \cup (2, \infty), x \in \mathbb{R} \} \)
      Decreasing: \( \{ x \mid x \in (-\infty, -2) \cup (0, 2), x \in \mathbb{R} \} \)
    1. \( y = \frac{1}{10} (x + 1)(3x - 2)(x - 3) \)
    2. \( \left(0, \frac{3}{5} \right) \)
      Sketch of y = (1/10)(x+1)(3x-2)(x-3)
    3. \( y = -\frac{1}{10} (x + 1)(3x - 2)(x - 3) \)
    4. \( y = -\frac{1}{10} (x - 1)(3x + 2)(x + 3) \)
  10. \( y = -\frac{1}{8}(x - 3)(x - 1)(x - 7) \).  The zeros of the new function are \(1, 3,\) and \(7\).
    1. \( \frac{17}{4} \)
    2. \( 0 \)

Finite Differences of Polynomial Functions


      1. Quadratic
      2. \( a = -3 \)
      1. Cubic
      2. \( a = 3 \)
      1. Quartic
      2. \( a = 2 \)
    1. The \( 5^{th} \) difference is constant.
    2. \( -480 \)
  3. Answers may vary.
    Quartic, negative leading coefficient, zeros at x=-4,-1; double zero at x=3
  4. \( y = -\frac{10}{3}x^2(x + 5) \)
  5. \( y = \frac{1}{2}\left (x - \frac{1}{2}\right )(x + 3)^3 \)
    1. \( \Delta^3 y = 12, y = 2x^3 - 3x^2 + 12x - 4 \)
    2. \( \Delta^3 y = -18, y = -3x^3 + 24x - 5 \)
    3. \( \Delta^3 y = -6, y = -x^3 + 3x^2 + 2 \)
    1. \( p(t) = t^3 - 27t^2 + 3t + 8538 \)
    2. \( 8030 \)
    3. \( p = 24536 \)
    4. Not realistic; \( p \rightarrow \infty \) as \( t \rightarrow \infty \)
  8. \( f(4) = 30,\ f(5) = 55,\ f(6) = 91,\ f(x) = \frac{1}{3}x^3 + \frac{1}{2}x^2 + \frac{1}{6}x \) or \( \dfrac{2x^3 + 3x^2 + x}{6} \)
  9. \( f(n) = \frac{1}{2}n^2 + \frac{1}{2}n + 1 \left(\text{or } = 1 + \dfrac{n(n+1)}{2} \right) \)

Even and Odd Polynomial Functions


      1. Odd degree
      2. Odd
      1. Odd degree
      2. Neither
      1. Odd degree
      2. Odd
      1. Even degree
      2. Even
      1. Even degree
      2. Even
      1. Odd degree
      2. Neither
      1. Even degree
      2. Neither
      1. Odd degree
      2. Odd
      1. Even degree
      2. Even
      1. Odd degree
      2. Neither
    1. Even
    2. Neither
    3. Odd
    1. True
    2. False
    3. False
    4. True
    5. False
    6. True
    1. Answers may vary.
      Sketch of y = -2x(x-4)^2(x+4)^2
    2. See solutions.
    3. Odd
  5. Answers may vary.
    1. Graph of y = -x^4 + 4x^2
    2. Graph of y = x^3 - 4x and Graph of y = x(x-2)^2(x+2)^2
    1. Answers may vary. One such equation is \( y = (x - 2)^2(x + 2)^2 \); note that \( f(-x) = (-x - 2)^2(-x + 2)^2 = (x + 2)^2(x - 2)^2 = f(x) \).
    2. Answers may vary. One such equation is \( y = x(x - 2)(x + 2) \).
      1. Not odd
      2. Odd
      3. Odd
      4. Odd
      5. Not odd
      6. Odd
      1. Not even
      2. Even
      3. Even
      4. Even
      5. Even
      6. Even