AP PreCalculus Flashcards: Polynomial Functions and Complex Zeros
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 16 cards to help you master important concepts.
How can you identify the key characteristics of a polynomial's zeros?
Key characteristics of zeros can be identified when suitable factorizations are available or by using technology to analyze the graph and values.
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How can you identify the key characteristics of a polynomial's zeros?
Key characteristics of zeros can be identified when suitable factorizations are available or by using technology to analyze the graph and values.
How many complex zeros does a polynomial function of degree n have?
A polynomial function of degree n has exactly n complex zeros, when counting for multiplicities.
If a polynomial has a known zero of 5 - 2i, what other number must also be a zero?
Based on the Complex Conjugate Root Theorem, its conjugate, 5 + 2i, must also be a zero of the polynomial.
What is the relationship between the signs of output values near a real zero with even multiplicity?
The signs of the output values are the same for input values on either side of a real zero with even multiplicity.
What is the Complex Conjugate Root Theorem for polynomials?
If a non-real complex number a+bi is a zero of a polynomial function p, then its conjugate a-bi must also be a zero of p.
What is the analytical property of an odd function?
An odd function analytically has the property f(-x) = -f(x) for all x in its domain.
How do you determine if a polynomial function is even or odd?
A function is determined to be even or odd by testing its analytical properties, such as f(-x) = f(x) for even or f(-x) = -f(x) for odd.
How can you determine the degree of a polynomial from a table of values?
The degree can be found by examining the successive differences of the output values over equal-interval input values until a constant difference is found.
For a polynomial, the output values for inputs near x=2 are both positive. What can you infer about the zero at x=2?
Since the signs of the output values are the same for inputs near x=2, the real zero at x=2 has an even multiplicity.
A function f(x) is symmetric about the point (0,0). What property must it satisfy?
Because it is an odd function, it must satisfy the property f(-x) = -f(x).
If a polynomial has only real coefficients, what is true about its non-real zeros?
If a polynomial with real coefficients has a non-real zero (a+bi), then its complex conjugate (a-bi) must also be a zero.
What is a zero of a polynomial function p?
A zero of a polynomial function p is a complex number 'a' for which p(a) = 0.
Describe the graphical symmetry of an odd function.
An odd function is graphically symmetric with respect to the origin, the point (0,0).
A polynomial function of degree 5 must have how many complex zeros?
The function must have exactly 5 complex zeros when counting multiplicities.
What is a root of the equation p(x) = 0?
A root of p(x) = 0 is a complex number 'a' that makes the equation true, meaning p(a) = 0.
What is the graphical effect of a real zero with even multiplicity?
If a real zero has even multiplicity, the signs of the output values are the same on both sides of the zero, meaning the graph touches the x-axis at that point but does not cross it.