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Math 30-1 · Worksheets

Polynomial, radical, rational · Set B

Ten questions of mixed difficulty, covering Polynomial, radical, rational. Print it, or work through it on screen — the answer key starts on its own page.

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Polynomial, radical, rational · Set B

Math 30-1 · maddyhelps.com

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  1. What does the graph of y = (x − 2) / ((x − 2)(x + 1)) have?

    1. a) a hole at x = −1 and a vertical asymptote at x = 2
    2. b) vertical asymptotes at x = 2 and x = −1
    3. c) a hole at x = 2 and a vertical asymptote at x = −1
    4. d) no asymptotes and no holes

  2. Which describes the end behaviour of y = x³?

    1. a) up to the left, down to the right
    2. b) down to the left, up to the right
    3. c) up on both ends
    4. d) down on both ends

  3. Which is a factor of P(x) = x³ − 7x + 6?

    1. a) x − 3
    2. b) x − 1
    3. c) x + 2
    4. d) x + 1

  4. What is the remainder when P(x) = x² + 3x − 1 is divided by x − 2?

    1. a) −11
    2. b) 1
    3. c) 9
    4. d) 3

  5. What is the vertical asymptote of y = 1/(x − 3)?

    1. a) x = −3
    2. b) y = 0
    3. c) y = 3
    4. d) x = 3

  6. What is the degree of y = 3x⁴ − 2x + 7?

    1. a) 3
    2. b) 4
    3. c) 1
    4. d) 7

  7. When graphing y = √f(x) from y = f(x), the invariant points occur where:

    1. a) the graph crosses the y-axis
    2. b) x = 0 and x = 1
    3. c) y = 0 and y = 1
    4. d) y = 1 only

  8. Which polynomial of least degree has a zero of −2 with multiplicity 2 and a zero of 3?

    1. a) y = (x − 2)²(x + 3)
    2. b) y = (x + 2)(x − 3)²
    3. c) y = (x + 2)²(x − 3)
    4. d) y = (x + 2)(x − 3)

  9. If P(4) = 0, which statement must be true?

    1. a) x + 4 is a factor of P(x)
    2. b) x − 4 is a factor of P(x)
    3. c) P(x) has degree 4
    4. d) 4 is the remainder when P(x) is divided by x

  10. What is the horizontal asymptote of y = (2x + 1)/(x − 3)?

    1. a) x = 3
    2. b) y = −1/3
    3. c) y = 2
    4. d) y = 0

Answer key · Polynomial, radical, rational · Set B

Math 30-1 · maddyhelps.com

  1. c) a hole at x = 2 and a vertical asymptote at x = −1 — x − 2 cancels from top and bottom, which leaves a hole (point of discontinuity) at x = 2. x + 1 does not cancel, so x = −1 is a vertical asymptote.
  2. b) down to the left, up to the right — Odd degree means the ends go opposite ways; a positive leading coefficient means it rises to the right. So it starts low and ends high.
  3. b) x − 1 — Test each: P(1) = 1 − 7 + 6 = 0, so x − 1 is a factor. P(−1), P(3) and P(−2) all give 12, not 0.
  4. c) 9 — The remainder theorem says the remainder is P(2): 4 + 6 − 1 = 9. No long division needed.
  5. d) x = 3 — The function is undefined where the denominator is zero: x − 3 = 0, so x = 3. y = 0 is the horizontal asymptote.
  6. b) 4 — The degree is the highest power of x, which is 4. The 3 is the leading coefficient and the 7 is the constant term.
  7. c) y = 0 and y = 1 — √0 = 0 and √1 = 1, so points with those y-values stay exactly where they are. Everywhere else the y-value changes.
  8. c) y = (x + 2)²(x − 3) — A zero of −2 comes from the factor x + 2, and multiplicity 2 squares it. A zero of 3 comes from x − 3. That gives degree 3.
  9. b) x − 4 is a factor of P(x) — The factor theorem: if P(a) = 0, then x − a is a factor. Here a = 4, so x − 4.
  10. c) y = 2 — The top and bottom have the same degree, so the asymptote is the ratio of leading coefficients: 2/1 = 2. x = 3 is the vertical asymptote.