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Math 10C

Math 10C · Worksheets

Mixed review · Numbers and polynomials

Ten questions of mixed difficulty, covering Numbers, roots and powers, Polynomials. Print it, or work through it on screen — the answer key starts on its own page.

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Mixed review · Numbers and polynomials

Math 10C · maddyhelps.com

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

  1. In simplest form, √48 is

    1. a) 2√12
    2. b) 16√3
    3. c) 4√12
    4. d) 4√3

  2. Factored, 3x² + 7x + 2 is

    1. a) 3(x + 1)(x + 2)
    2. b) (3x + 2)(x + 1)
    3. c) (3x + 1)(x + 2)
    4. d) (3x + 7)(x + 2)

  3. The value of 16^(1/2) is

    1. a) 32
    2. b) 8
    3. c) 4
    4. d) 2

  4. Factored, x² − 25 is

    1. a) (x − 5)(x + 5)
    2. b) x(x − 25)
    3. c) (x − 25)(x + 1)
    4. d) (x − 5)²

  5. Factored completely, 2x² + 10x + 12 is

    1. a) (2x + 4)(x + 3)
    2. b) 2(x + 2)(x + 3)
    3. c) 2(x + 1)(x + 6)
    4. d) (x + 2)(x + 6)

  6. x⁵ × x³ equals

    1. a) x⁸
    2. b) 2x⁸
    3. c) x¹⁵
    4. d) x²

  7. Expanded, (x + 2)(x + 5) is

    1. a) x² + 7x + 7
    2. b) x² + 10x + 7
    3. c) x² + 7x + 10
    4. d) x² + 10

  8. The sum of a rational number and an irrational number is

    1. a) always irrational
    2. b) always a whole number
    3. c) always rational
    4. d) sometimes rational

  9. Expanded, 3(x + 4) is

    1. a) 3x + 4
    2. b) 3x·4
    3. c) x + 12
    4. d) 3x + 12

  10. Factored, x² + 9x + 20 is

    1. a) (x − 4)(x − 5)
    2. b) (x + 4)(x + 5)
    3. c) (x + 2)(x + 10)
    4. d) (x + 1)(x + 20)

Answer key · Mixed review · Numbers and polynomials

Math 10C · maddyhelps.com

  1. d) 4√3 — Pull out the largest perfect square factor: 48 = 16 × 3, so √48 = 4√3. The option 2√12 is equal in value but not simplified, because 12 still contains the perfect square 4.
  2. c) (3x + 1)(x + 2) — Check by expanding: (3x + 1)(x + 2) gives 3x² + 6x + x + 2 = 3x² + 7x + 2 ✓. The near-miss (3x + 2)(x + 1) gives a middle term of 5x, so which bracket gets the 2 matters.
  3. c) 4 — A fractional exponent is a root: the denominator says which root. So 16^(1/2) = √16 = 4. And 16^(3/2) would be (√16)³ = 64 — take the root first, then the power, because the numbers stay smaller.
  4. a) (x − 5)(x + 5) — A difference of squares always splits into a difference times a sum. Note that a <i>sum</i> of squares like x² + 25 does not factor over the real numbers at all.
  5. b) 2(x + 2)(x + 3) — Take out the common factor 2 first, leaving x² + 5x + 6 = (x + 2)(x + 3). The option (2x + 4)(x + 3) expands correctly but is not <i>completely</i> factored, because 2x + 4 still has a common factor.
  6. a) x⁸ — Multiplying powers of the same base adds the exponents, because you are just counting how many copies of x there are in total: five plus three is eight. The bases do not multiply.
  7. c) x² + 7x + 10 — Every term in the first bracket multiplies every term in the second: x² + 5x + 2x + 10, then collect to x² + 7x + 10. Checking with x = 1 gives 3 × 6 = 18 on the left and 1 + 7 + 10 = 18 on the right ✓.
  8. a) always irrational — If the sum were rational, subtracting the rational part would make the irrational number rational too, which is a contradiction. Note the contrast: the sum of two <i>irrational</i> numbers can be rational, as √2 + (−√2) shows.
  9. d) 3x + 12 — Distribute the 3 over both terms in the bracket. Multiplying only the first term is the error that produces 3x + 4, and it is worth checking every expansion by substituting a number.
  10. b) (x + 4)(x + 5) — Find two numbers multiplying to 20 and adding to 9: 4 and 5. The pairs 2 and 10 or 1 and 20 both multiply to 20 but add to 12 and 21, so they fail the second test.