Polynomial and Rational Functions

Grade 11 · Math · Study Guide

A deep dive into the behavior of polynomial and rational functions including end behavior, zeros, and asymptotes.

A polynomial function is a sum of terms of the form ax^n where n is a non-negative integer. The degree of a polynomial is the highest power present. End behavior is determined by the leading term: if the leading coefficient is positive and degree is even, both ends go up; if odd, the left end goes down and right end goes up. The Fundamental Theorem of Algebra states a degree-n polynomial has exactly n complex zeros (counting multiplicity).

The Rational Root Theorem states that if p/q is a rational zero of a polynomial with integer coefficients, then p divides the constant term and q divides the leading coefficient. Synthetic division efficiently evaluates polynomials and checks for zeros. Multiplicity affects behavior at zeros: an even multiplicity zero means the graph touches but does not cross the x-axis; odd multiplicity means the graph crosses.

Rational functions are ratios of polynomials: f(x) = p(x)/q(x). They have vertical asymptotes where q(x) = 0 (and p(x) ≠ 0), and horizontal or oblique asymptotes determined by comparing degrees. If degree of p < degree of q, horizontal asymptote is y = 0. If degrees are equal, y = leading coefficient ratio. If degree of p > degree of q by 1, there is an oblique asymptote found by polynomial long division.

Key terms

Degree: The highest power of the variable in a polynomial; determines end behavior and maximum number of zeros.
Multiplicity: The number of times a root appears as a factor; even multiplicity means the graph touches but doesn't cross the x-axis.
Vertical Asymptote: A vertical line x = a where a rational function is undefined and the output approaches ±∞.
Horizontal Asymptote: A horizontal line y = b that the graph of a rational function approaches as x → ±∞.
Rational Root Theorem: Possible rational zeros of a polynomial are ±(factors of constant term)/(factors of leading coefficient).
Oblique Asymptote: A slanted asymptote that occurs when the degree of the numerator exceeds the degree of the denominator by exactly 1.

Flashcards

End behavior rule (even degree, positive leading coeff): Both ends of the graph go up (↑ ↑)
End behavior rule (odd degree, positive leading coeff): Left end goes down, right end goes up (↓ ↑)
Vertical asymptotes of rational function: Set denominator = 0 and check that numerator ≠ 0 at those x-values
Horizontal asymptote comparison: deg(p) < deg(q): y=0; deg(p)=deg(q): y=a/b; deg(p)>deg(q) by 1: oblique
Rational Root Theorem: Possible rational roots = ±factors(constant) / factors(leading coefficient)
Multiplicity even: Graph touches x-axis and bounces back (does not cross)
Multiplicity odd: Graph crosses x-axis at that zero
Hole in a rational function: Occurs when a factor cancels from both numerator and denominator; graph has a removable discontinuity

Questions

What is the end behavior of f(x) = -3x^4 + 2x² - 1?
1. Up on both ends
2. Down on both ends
3. Down left, up right
4. Up left, down right
Show answer
Answer: Down on both ends
The leading term is -3x^4: negative leading coefficient with even degree → both ends go down.
The function f(x) = (x² - 4)/(x - 2) has a vertical asymptote at x = 2.
Show answer
Answer: False
x² - 4 = (x-2)(x+2), so the (x-2) cancels. There is a hole at x=2, not a vertical asymptote.
A polynomial of degree 5 has at most ___ real zeros.
Show answer
Answer: 5
By the Fundamental Theorem of Algebra, a degree-5 polynomial has exactly 5 complex zeros, so at most 5 real zeros.
What is the horizontal asymptote of f(x) = (3x² + 1)/(x² - 2)?
1. y = 0
2. y = 1
3. y = 3
4. No horizontal asymptote
Show answer
Answer: y = 3
Both numerator and denominator have degree 2; the horizontal asymptote is the ratio of leading coefficients: y = 3/1 = 3.
If x = 3 is a zero of multiplicity 2 for a polynomial, the graph crosses the x-axis at x = 3.
Show answer
Answer: False
Even multiplicity means the graph touches but does not cross the x-axis at that zero.

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