Algebra 2 Practice Test: Polynomial Zeros, Multiplicity and End Behavior
- Polynomial Zeros and End Behavior
- 10 questions
- 13 minutes
These 10 questions connect a polynomial rule to the shape of its graph. Tip: the leading term decides the end behavior, and the multiplicity of a zero decides whether the graph crosses or touches.
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?
Correct answer: A
The leading term −2x5 has odd degree and a negative coefficient, so the graph goes up on the left and down on the right.
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Question 2
Which describes the end behavior of f(x) = 3x4 − 5x3 + 2?
Correct answer: C
The degree is even and the leading coefficient 3 is positive, so both ends point up, like a parabola that opens upward.
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Question 3
At which zero does the graph of f(x) = x(x − 2)2(x + 5) touch the x-axis and turn around?
Correct answer: D
The factor (x − 2)2 gives the zero 2 an even multiplicity, so the graph touches the axis there without crossing. The zeros 0 and −5 have multiplicity 1, so the graph crosses.
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Question 4
Which polynomial has zeros 1, −3 and 4 and a leading coefficient of 1?
Correct answer: B
A zero r gives the factor (x − r). For −3 that is (x + 3). The first option has every sign reversed and would have zeros −1, 3 and −4.
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Question 5
What is the greatest number of turning points the graph of a degree 5 polynomial can have?
Correct answer: D
A polynomial of degree n has at most n − 1 turning points, so a degree 5 polynomial has at most 4.
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Question 6
By the Rational Root Theorem, which number is not a possible rational zero of 2x3 − 5x2 + x − 3?
Correct answer: D
Possible rational zeros are (factors of 3) ÷ (factors of 2): ±1, ±3, ±12, ±32. The number 23 would need 2 to divide the constant and 3 to divide the leading coefficient, which is reversed.
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Question 7
One zero of f(x) = x3 − 4x2 + x + 6 is 3. What are the other zeros?
Correct answer: A
Divide by (x − 3): f(x) = (x − 3)(x2 − x − 2) = (x − 3)(x − 2)(x + 1). The other zeros are 2 and −1.
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Question 8
A polynomial with real coefficients has zeros 2 and 3 + i. Which number must also be a zero?
Correct answer: B
Non-real zeros of a polynomial with real coefficients come in conjugate pairs, so 3 − i is also a zero. Nothing forces −2 to be a zero.
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Question 9
Counting multiplicity and including complex zeros, how many zeros does a polynomial of degree 6 have?
Correct answer: C
By the Fundamental Theorem of Algebra, a degree n polynomial has exactly n complex zeros when repeated zeros are counted by multiplicity.
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Question 10
What does the graph of f(x) = (x + 1)3(x − 4) do at x = −1?
Correct answer: B
The zero −1 has odd multiplicity 3, so the graph crosses the axis, and because the multiplicity is greater than 1 it flattens out, like y = x3 at the origin.
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