Algebra 2 Practice Test: Polynomial and Rational Functions
- Polynomial and Rational Functions
- 15 questions
- 20 minutes
These 15 questions practice polynomial division, the remainder and factor theorems, zeros and multiplicity, end behavior, asymptotes and holes, simplifying and adding rational expressions and solving rational equations. Always check solutions of rational equations: a value that makes a denominator 0 is not a solution.
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Question 1
What is the quotient when x3 − 4x2 + x + 6 is divided by x − 2?
Correct answer: A
Synthetic division with 2 on the coefficients 1, −4, 1, 6 gives 1, −2, −3 with remainder 0. Dividing by −2 instead of 2 leads to x2 − 6x + 13 with a nonzero remainder.
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Question 2
What is the remainder when p(x) = 2x3 + 3x2 − 5x + 4 is divided by x + 2?
Correct answer: B
By the remainder theorem the remainder is p(−2) = −16 + 12 + 10 + 4 = 10. The value 22 is p(2), which belongs to division by x − 2.
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Question 3
If x − 3 is a factor of x3 − 2x2 + kx + 6, what is the value of k?
Correct answer: D
By the factor theorem, p(3) = 0: 27 − 18 + 3k + 6 = 0, so 15 + 3k = 0 and k = −5.
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Question 4
What are the zeros of f(x) = x3 − x2 − 12x?
Correct answer: A
Factor out x: x(x2 − x − 12) = x(x − 4)(x + 3). The common factor x gives the zero 0, which is easy to lose.
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Question 5
Which describes the end behavior of f(x) = −2x5 + 3x2 − 1?
Correct answer: C
The leading term −2x5 has odd degree and a negative coefficient, so the graph falls to the right and rises to the left. Both ends point the same way only for even degree.
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Question 6
The function f(x) = (x − 1)2(x + 4)3 is graphed. Which statement about the graph is true?
Correct answer: B
The zero 1 has even multiplicity (2), so the graph touches and turns there. The zero −4 has odd multiplicity (3), so the graph crosses the axis there.
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Question 7
A polynomial function of degree 3 with real coefficients has zeros 4 and 2 − i. Which number must also be a zero?
Correct answer: D
Non-real zeros of a polynomial with real coefficients come in conjugate pairs, so 2 + i is also a zero. That makes three zeros, which matches degree 3.
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Question 8
What are the vertical asymptotes of f(x) = x + 2x2 − 9?
Correct answer: C
The denominator x2 − 9 = (x − 3)(x + 3) is 0 at x = 3 and x = −3, and the numerator is not 0 there. x = −2 makes the function equal 0, so it is an x-intercept.
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Question 9
What is the horizontal asymptote of f(x) = 4x2 − 12x2 + 5x?
Correct answer: B
The degrees of numerator and denominator are equal, so the asymptote is the ratio of leading coefficients: y = 42 = 2. y = 0 applies only when the denominator has the higher degree.
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Question 10
Which statement describes the graph of f(x) = x2 − 4x − 2?
Correct answer: D
(x − 2)(x + 2)x − 2 = x + 2 for x ≠ 2. The common factor cancels, so the graph is the line y = x + 2 with a hole where x = 2, at height 2 + 2 = 4.
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Question 11
Simplify x2 − x − 6x2 − 9 for x ≠ ±3.
Correct answer: C
(x − 3)(x + 2)(x − 3)(x + 3) = x + 2x + 3. You may cancel only common factors, not individual terms such as x2.
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Question 12
What is the solution of xx − 3 = 3x − 3 + 4?
Correct answer: C
Multiplying by x − 3 gives x = 3 + 4(x − 3), so x = 3. But x = 3 makes the denominators 0, so it is extraneous and the equation has no solution.
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Question 13
Which expression is equivalent to 2x + 3x + 1?
Correct answer: B
Use the common denominator x(x + 1): 2(x + 1) + 3xx(x + 1) = 5x + 2x(x + 1). Adding numerators and denominators directly gives 52x + 1, which is not equivalent.
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Question 14
Which expression is equivalent to (x + 2)3?
Correct answer: D
(a + b)3 = a3 + 3a2b + 3ab2 + b3 = x3 + 3(x2)(2) + 3(x)(4) + 8 = x3 + 6x2 + 12x + 8. Cubing each term separately gives x3 + 8.
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Question 15
Pipe A can fill a tank in 6 hours, and pipe B can fill it in 3 hours. How many hours does it take to fill the empty tank with both pipes open?
Correct answer: A
Together the pipes fill 16 + 13 = 12 of the tank per hour, so they need 2 hours. 4.5 hours is the average of the two times, which ignores that they work at the same time.
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