Polynomial and Rational Functions Quiz

Grade 11 · Math · Quiz

Assess your skills in analyzing polynomial behavior, rational function asymptotes, and zeros.

Questions

1. Which of the following is a possible rational zero of f(x) = 2x³ - 3x + 6?

   1. ±4
   2. ±3/2
   3. ±4/3
   4. ±5
   Show answer

   Answer: ±3/2

   Possible rational zeros = ±(factors of 6)/(factors of 2) = ±1, ±2, ±3, ±6, ±1/2, ±3/2. Only ±3/2 is listed.

2. What are the vertical asymptotes of f(x) = (x+1)/((x-3)(x+2))?

   1. x = -1
   2. x = 3 and x = -2
   3. x = 3 only
   4. x = -2 only
   Show answer

   Answer: x = 3 and x = -2

   Vertical asymptotes occur where the denominator is zero (and numerator ≠ 0): x = 3 and x = -2.

3. The graph of y = (x-2)³(x+1)² crosses the x-axis at x =

   1. x = 2 only
   2. x = -1 only
   3. x = 2 and x = -1, crossing both
   4. x = 2 (crosses) and x = -1 (touches)
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   Answer: x = 2 (crosses) and x = -1 (touches)

   (x-2)³ has odd multiplicity → crosses at x=2; (x+1)² has even multiplicity → touches (bounces) at x=-1.

4. What is the oblique asymptote of f(x) = (x² + 2x + 3)/(x - 1)?

   1. y = x + 1
   2. y = x + 3
   3. y = x + 2
   4. y = x
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   Answer: y = x + 3

   Dividing: x² + 2x + 3 ÷ (x-1) = x + 3 remainder 6. Oblique asymptote: y = x + 3.

5. How many complex zeros does f(x) = x^5 - 3x + 1 have?

   1. 3
   2. 4
   3. 5
   4. It depends on the coefficients
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   Answer: 5

   By the Fundamental Theorem of Algebra, every degree-5 polynomial has exactly 5 complex zeros (counting multiplicity).

6. Which function has NO horizontal asymptote?

   1. f(x) = (2x+1)/(x-3)
   2. f(x) = (x²+1)/(x-1)
   3. f(x) = 3/(x²+1)
   4. f(x) = (x+2)/(x²+1)
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   Answer: f(x) = (x²+1)/(x-1)

   When degree(numerator) > degree(denominator) by more than 1, there is no horizontal asymptote (there is an oblique or higher-degree asymptote).

7. A polynomial with real coefficients can have exactly one non-real complex zero.

   Show answer

   Answer: False

   Non-real complex zeros of polynomials with real coefficients come in conjugate pairs. You cannot have exactly one.

8. The graph of y = (x²-9)/(x+3) is a line with a hole at x = ___.

   Show answer

   Answer: -3

   x²-9 = (x-3)(x+3); cancels (x+3) leaving y = x-3, a line with a hole at x = -3.

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