Precalculus Practice Test: Polynomial Functions
- Polynomial Functions
- 10 questions
- 15 minutes
These 10 questions cover end behavior, zeros and their multiplicity, the Remainder and Factor Theorems, and the Rational Zero Theorem. For end behavior, look only at the leading term: its sign and whether the degree is odd or even.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
Which describes the end behavior of f(x) = −2x5 + 3x2 − 1?
Correct answer: D
The leading term −2x5 has odd degree and a negative coefficient, so f(x) → −∞ as x → ∞ and f(x) → ∞ as x → −∞. Falling on the left and rising on the right is the pattern for a positive leading coefficient.
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Question 2
The graph of f(x) = (x − 3)2(x + 1)3 touches the x-axis and turns around at one zero. Which zero is it?
Correct answer: A
The factor (x − 3) has even multiplicity 2, so the graph touches the axis at x = 3 and turns back. At x = −1 the multiplicity is 3, which is odd, so the graph crosses there.
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Question 3
What is the remainder when x3 − 4x2 + 2x + 5 is divided by x − 3?
Correct answer: B
By the Remainder Theorem the remainder is f(3) = 27 − 36 + 6 + 5 = 2. The value −64 is f(−3), which comes from using the wrong sign of the divisor.
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Question 4
A cubic polynomial has leading coefficient 1 and zeros 2, −1 and 4. What is its constant term?
Correct answer: C
The polynomial is (x − 2)(x + 1)(x − 4), and the constant term is (−2)(1)(−4) = 8. The value 5 is the sum of the zeros, which is related to the x2 coefficient, not the constant term.
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Question 5
A polynomial with real coefficients has zeros 3i and 2. Which number must also be a zero?
Correct answer: D
Complex zeros of a polynomial with real coefficients come in conjugate pairs, so −3i must also be a zero. Nothing requires −2 to be a zero: real zeros do not come in pairs.
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Question 6
What is the greatest number of turning points the graph of a polynomial of degree 6 can have?
Correct answer: A
A polynomial of degree n has at most n − 1 turning points, so a degree 6 polynomial has at most 5. The number 6 is the greatest possible number of real zeros, not turning points.
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Question 7
For what value of k is x + 2 a factor of x3 + kx2 − 4?
Correct answer: B
By the Factor Theorem, f(−2) = 0: −8 + 4k − 4 = 0, so k = 3. The value −1 comes from substituting x = 2 instead of x = −2.
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Question 8
Which number is NOT a possible rational zero of f(x) = 2x3 − x + 6, according to the Rational Zero Theorem?
Correct answer: C
Possible rational zeros are ±p/q, where p divides 6 and q divides 2. A denominator of 3 is impossible because 3 does not divide the leading coefficient 2.
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Question 9
What are all the solutions of x3 − 6x2 + 8x = 0?
Correct answer: A
Factor out x: x(x2 − 6x + 8) = x(x − 2)(x − 4) = 0, so x = 0, 2 or 4. Dividing both sides by x would lose the solution x = 0.
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Question 10
What is the sum of the zeros (counted with multiplicity, including complex zeros) of f(x) = 3x3 − 12x2 + x − 7?
Correct answer: D
For ax3 + bx2 + cx + d, the zeros add up to −b/a = 12/3 = 4. The answer −4 keeps the sign of b instead of taking its opposite.
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