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Math 30-1 · Worksheets

Polynomial, radical, rational · Set A

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 A

Math 30-1 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. When x³ + kx² − 4 is divided by x − 2, the remainder is 8. What is k?

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

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

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

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

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

  4. What are the zeros of y = (x − 2)(x + 5)?

    1. a) 2 and 5
    2. b) −2 and 5
    3. c) −10
    4. d) 2 and −5

  5. What is the domain of y = √x?

    1. a) x ≤ 0
    2. b) all real numbers
    3. c) x ≥ 0
    4. d) x > 0

  6. At x = 1, the graph of y = (x − 1)²(x + 3):

    1. a) has a hole
    2. b) touches the x-axis and turns around
    3. c) crosses the x-axis
    4. d) has a vertical asymptote

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

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

  8. By the integral zero theorem, which are the possible integer zeros of P(x) = x³ − 2x² − 5x + 6?

    1. a) ±1, ±2, ±3, ±6
    2. b) ±1, ±5
    3. c) ±1, ±2, ±5
    4. d) 1, 2, 3 and 6 only

  9. What is the horizontal asymptote of y = 1/x?

    1. a) y = 0
    2. b) y = 1
    3. c) there is none
    4. d) x = 0

  10. What is the range of y = √(x − 2) + 3?

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

Answer key · Polynomial, radical, rational · Set A

Math 30-1 · maddyhelps.com

  1. d) 1 — The remainder is P(2): 8 + 4k − 4 = 8. So 4 + 4k = 8, and k = 1.
  2. c) 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.
  3. a) 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. d) 2 and −5 — Set each factor to zero: x − 2 = 0 gives 2 and x + 5 = 0 gives −5. The signs flip from what you see in the brackets.
  5. c) x ≥ 0 — You cannot take the square root of a negative number, but √0 = 0 is fine, so x ≥ 0.
  6. b) touches the x-axis and turns around — The factor x − 1 is squared, so x = 1 is a zero of multiplicity 2. Even multiplicity touches and bounces back; odd multiplicity crosses.
  7. 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.
  8. a) ±1, ±2, ±3, ±6 — Integer zeros must divide the constant term, 6, and can be positive or negative: ±1, ±2, ±3, ±6. (1, −2 and 3 turn out to work.)
  9. a) y = 0 — As x gets very large or very negative, 1/x gets closer and closer to 0 without reaching it. x = 0 is the vertical asymptote.
  10. b) y ≥ 3 — √ is never negative, so the smallest value of the root is 0. Add 3 and the smallest y is 3. The −2 inside affects the domain, not the range.