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Math 9

Math 9 · Worksheets

Polynomials

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

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Polynomials

Math 9 · maddyhelps.com

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  1. The opposite of 2x² − 3x + 1 is

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

  2. 3x(2x − 5) expands to

    1. a) 6x² − 5
    2. b) 6x² − 15
    3. c) 6x² − 15x
    4. d) 5x² − 15x

  3. −2x(3x − 4) − 5x simplifies to

    1. a) −6x² − 5x − 4
    2. b) −6x² − 13x
    3. c) −6x² + 13x
    4. d) −6x² + 3x

  4. Which of these is a trinomial of degree 2 once it is simplified?

    1. a) x² + 2x − x² + 5
    2. b) 5x² + 2x² − 3
    3. c) 4 − x + 2x² − 1
    4. d) 3x² − x + x

  5. A student models a polynomial with algebra tiles: two positive x²-tiles, three negative x-tiles and four positive unit tiles. Which polynomial is it?

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

  6. A rectangular garden is 2x metres wide and 3x + 4 metres long. Which expression gives its area in square metres?

    1. a) 10x + 8
    2. b) 6x² + 4
    3. c) 6x² + 8x
    4. d) 5x + 4

  7. The degree of the polynomial 4x² − 3x + 7 is

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

  8. A rectangle has length 3x + 2 and width x − 1. Which expression is its perimeter?

    1. a) 8x + 6
    2. b) 4x + 1
    3. c) 8x + 1
    4. d) 8x + 2

  9. In the polynomial 6x² − x + 9, the coefficient of x is

    1. a) 6
    2. b) −1
    3. c) 1
    4. d) 0

  10. (10x² + 15x) ÷ 5x simplifies to

    1. a) 2x² + 3x
    2. b) 2x + 3x
    3. c) 2x + 15x
    4. d) 2x + 3

Answer key · Polynomials

Math 9 · maddyhelps.com

  1. d) −2x² + 3x − 1 — The opposite changes the sign of every term, like flipping every algebra tile over: −2x² + 3x − 1. Changing only the first sign gives −2x² − 3x + 1, and that is exactly the error that ruins polynomial subtraction.
  2. c) 6x² − 15x — Multiply 3x by each term: 3x × 2x = 6x² and 3x × (−5) = −15x. Multiplying only the first term gives 6x² − 5, and 6x² − 15 forgets that the x multiplies the second term too.
  3. d) −6x² + 3x — −2x × 3x = −6x², and −2x × (−4) = +8x because a negative times a negative is positive. Then 8x − 5x = 3x, giving −6x² + 3x. Writing −8x instead of +8x leads to −6x² − 13x.
  4. c) 4 − x + 2x² − 1 — Collect like terms first: 4 − x + 2x² − 1 = 2x² − x + 3, which has three terms and highest exponent 2. The others look like three or four terms but shrink: 3x² − x + x = 3x², x² + 2x − x² + 5 = 2x + 5, and 5x² + 2x² − 3 = 7x² − 3.
  5. c) 2x² − 3x + 4 — Each tile shape is one kind of term: big squares are x², rectangles are x and small squares are 1. Two positive x², three negative x and four positive units make 2x² − 3x + 4; ignoring the colour of the negative tiles gives 2x² + 3x + 4.
  6. c) 6x² + 8x — Area is width × length: 2x(3x + 4) = 6x² + 8x, with the 2x multiplying both terms. 10x + 8 is the perimeter, 2(2x) + 2(3x + 4), and 5x + 4 just adds the two sides.
  7. b) 2 — The degree is the highest exponent on the variable: the 2 in x². There are 3 terms, and 4 is a coefficient — neither of those is the degree.
  8. d) 8x + 2 — Perimeter is all four sides: 2(3x + 2) + 2(x − 1) = 6x + 4 + 2x − 2 = 8x + 2. Adding the length and width once gives 4x + 1, which is only half the way around, and 8x + 6 treats the − 1 as + 1.
  9. b) −1 — −x means −1 × x, so the coefficient is −1: the sign in front belongs to the term. A variable with no number written in front has a coefficient of 1 or −1, never 0.
  10. d) 2x + 3 — Divide each term by 5x: 10x² ÷ 5x = 2x and 15x ÷ 5x = 3. Dividing only the numbers gives 2x² + 3x, and dividing only the first term leaves 15x behind. Check by multiplying back: 5x(2x + 3) = 10x² + 15x ✓.