Polynomial Functions Quiz

Grade 11 · Math · Quiz

10 questions on polynomial degree, zeros, factoring, and graph behavior.

Questions

1. How many complex zeros (counting multiplicity) does a degree-6 polynomial have?

   1. 3
   2. 4
   3. 6
   4. 12
   Show answer

   Answer: 6

   By the Fundamental Theorem of Algebra, a degree-n polynomial has exactly n complex zeros.

2. What is the remainder when f(x) = x³ + 2x − 5 is divided by (x − 1)?

   1. −2
   2. 0
   3. −3
   4. 3
   Show answer

   Answer: −2

   Remainder Theorem: f(1) = 1 + 2 − 5 = −2.

3. Which is a zero of f(x) = 2x³ − 3x² − 11x + 6?

   1. x = 1
   2. x = 3
   3. x = −1
   4. x = 4
   Show answer

   Answer: x = 3

   f(3) = 2(27) − 3(9) − 11(3) + 6 = 54 − 27 − 33 + 6 = 0.

4. f(x) = −x³ + 2x + 1. As x → +∞, f(x) →?

   1. +∞
   2. 1
   3. −∞
   4. 0
   Show answer

   Answer: −∞

   Odd degree, negative leading coefficient → right end goes to −∞.

5. How many turning points can a degree-5 polynomial have at most?

   1. 4
   2. 5
   3. 6
   4. 3
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   Answer: 4

   A degree-n polynomial has at most n − 1 turning points; 5 − 1 = 4.

6. Factor completely: x³ − 8.

   1. (x − 2)(x² + 2x + 4)
   2. (x − 2)(x + 2)²
   3. (x − 2)³
   4. (x + 2)(x² − 2x + 4)
   Show answer

   Answer: (x − 2)(x² + 2x + 4)

   Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²) with a=x, b=2.

7. The graph of y = (x + 1)³ crosses the x-axis at x = −1 with what behavior?

   1. Touches and turns
   2. Crosses straight through
   3. Has an asymptote
   4. Is undefined
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   Answer: Crosses straight through

   Odd multiplicity (3) means the graph crosses the x-axis.

8. What is the maximum number of real zeros of a degree-4 polynomial?

   1. 2
   2. 3
   3. 4
   4. 8
   Show answer

   Answer: 4

   A degree-4 polynomial can have at most 4 real zeros.

9. Which polynomial has zeros at x = −3, 0, and 2?

   1. x(x − 3)(x + 2)
   2. x(x + 3)(x − 2)
   3. x(x − 3)(x − 2)
   4. (x + 3)(x − 2)
   Show answer

   Answer: x(x + 3)(x − 2)

   Set each factor to zero: x=0, x+3=0 → x=−3, x−2=0 → x=2. ✓

10. Divide f(x) = x² − 5x + 6 by (x − 2). What is the quotient?

    1. x − 3
    2. x + 3
    3. x − 2
    4. x − 1
    Show answer

    Answer: x − 3

    (x² − 5x + 6) ÷ (x − 2) = (x − 2)(x − 3) ÷ (x − 2) = x − 3.

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