Polynomial and Rational Functions flashcards

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What is degree and leading term?Polynomial Functions and End Behavior: The highest nonzero exponent and its coefficient.
Why ignore lower terms in end behavior?Polynomial Functions and End Behavior: Their relative influence shrinks as |x| grows.
What is even and odd degree?Polynomial Functions and End Behavior: The parity pattern of polynomial end behavior.
Can lower-degree terms change the end directions?Polynomial Functions and End Behavior: No, the leading term controls them.
What is interpreting graphs?Polynomial Functions and End Behavior: Relating degree to possible graph features.
Can a degree-four polynomial have five distinct real zeros?Polynomial Functions and End Behavior: No, it has at most four distinct zeros.
What is zeros as factors?Zeros, Multiplicity, and Graph Shape: The connection between polynomial zeros and factors.
How can a proposed zero be checked?Zeros, Multiplicity, and Graph Shape: Evaluate the polynomial at that input.
What is even multiplicity?Zeros, Multiplicity, and Graph Shape: A repeated root whose exponent is even.
Does an even-multiplicity root necessarily cross?Zeros, Multiplicity, and Graph Shape: No, it generally touches and turns locally.
What is odd multiplicity?Zeros, Multiplicity, and Graph Shape: A root repeated an odd number of times.
What does multiplicity count?Zeros, Multiplicity, and Graph Shape: The exponent on the associated linear factor.
What is remainder theorem?Factor and Remainder Theorems: A link between evaluation and division remainder.
What input is used for divisor x−r?Factor and Remainder Theorems: Evaluate at x=r.
What is factor theorem?Factor and Remainder Theorems: The zero-remainder criterion for a factor.
How can a false candidate be rejected quickly?Factor and Remainder Theorems: Evaluate p(r) and see whether it is nonzero.
What is factoring after division?Factor and Remainder Theorems: Reducing polynomial degree by a known factor.
Why compare root multiplicities with degree?Factor and Remainder Theorems: The total multiplicity of factors should match the degree over complex numbers.
What is vertical asymptotes?Graphing Rational Functions: A vertical line approached near a nonremovable pole.
How is a hole different?Graphing Rational Functions: A hole comes from a cancelled common factor.
What is horizontal asymptotes?Graphing Rational Functions: A horizontal line approached at extreme inputs.
Must a graph avoid crossing a horizontal asymptote?Graphing Rational Functions: No, other rational functions can cross one.
What is holes and intercepts?Graphing Rational Functions: Valid intercepts and removable discontinuities.
Can an excluded x be an x-intercept?Graphing Rational Functions: No, the function has no output there.