mathematicspolynomialsrational functionsactive lesson

Active Lesson: Polynomials and Rational Functions

Mathimatikos TeamJune 8, 202610 min

Noen tilleggstekster i denne delen er foreløpig tilgjengelige på engelsk.

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Before You Start

Powers and exponents
Factoring basics
Reading intercepts from graphs

Lesson time

22 min

Objectives

What You Will Learn

Distinguish power, polynomial, and rational functions.
Use degree, leading coefficient, zeros, and factors to predict graph behavior.
Identify vertical and horizontal asymptotes in simple rational functions.

Activity

Preview Activity: Read the Shape Before the Formula

A graph can reveal algebraic structure before you know the exact formula.

If a graph crosses the $x$ -axis at $x = - 2$ , $x = 1$ , and $x = 3$ , what factors might appear?
If both ends of the graph point upward, should the degree be even or odd?
If a graph touches the $x$ -axis at $x = 1$ but does not cross, what might be true about the factor $x - 1$ ?

Hint

Crossing usually suggests an odd multiplicity. Touching and turning usually suggests an even multiplicity.

Solution

Possible factors are $(x + 2)$ , $(x - 1)$ , and $(x - 3)$ .

Both ends in the same direction suggest even degree.

Touching at $x = 1$ suggests $(x - 1)$ appears an even number of times.

Definition

Power and Polynomial Functions

p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}

A power function has the form $f (x) = k x^{n}$ .

A polynomial is a sum of power functions with nonnegative whole-number exponents.

The degree and leading coefficient usually control the far-left and far-right behavior.

Zeros and Turning Behavior

The polynomial $y = (x + 2) (x - 1)^{2} (x - 3)$ crosses at $x = - 2$ and $x = 3$ , but touches at $x = 1$ .

Interaktiv graf

Definition

Zeros and Factors

If $p (a) = 0$ , then $a$ is a zero of the polynomial.

A zero at $x = a$ usually corresponds to a factor $(x - a)$ .

A polynomial of degree $n$ has at most $n$ real zeros and at most $n - 1$ turning points.

Worked example

Worked Example: Build a Polynomial from Zeros

Build a polynomial with zeros $- 1$ , $2$ , and $2$ , and with positive leading coefficient.

Solution

The zero $- 1$ gives factor $(x + 1)$ .

The repeated zero $2$ gives factor $(x - 2)^{2}$ .

One simple answer is $p (x) = (x + 1) (x - 2)^{2}$ .

Definition

Rational Functions and Asymptotes

r (x) = \frac{P ( x )}{Q ( x )}

A rational function is a quotient of two polynomials.

The denominator cannot be zero, so zeros of $Q (x)$ are excluded from the domain.

A vertical asymptote can occur where the denominator is zero and the factor does not cancel.

Horizontal behavior depends on the degrees of $P (x)$ and $Q (x)$ .

Activity

Activity: Asymptote or Removable Hole?

Simplify $r (x) = \frac{x ^{2} - 1}{x - 1}$ for $x \neq = 1$ .
Does $x = 1$ create a vertical asymptote or a removable hole?
Compare with $s (x) = \frac{x + 1}{x - 1}$ . What happens near $x = 1$ ?

Hint

Factor $x^{2} - 1$ first.

Solution

$x^{2} - 1 = (x - 1) (x + 1)$ , so $r (x) = x + 1$ when $x \neq = 1$ .

$x = 1$ is a removable hole, not a vertical asymptote.

For $s (x)$ the factor does not cancel, so $x = 1$ is a vertical asymptote.

Summary

Power functions are the building blocks; polynomials are sums of those blocks.
Zeros and factors explain where a graph crosses or touches the $x$ -axis.
Rational functions require domain checks because division by zero controls holes and asymptotes.

Practice

Try These

Predict the end behavior of $p (x) = - 2 x^{5} + 3 x^{2} - 1$ .
Build a polynomial with zeros $0$ , $3$ , and $3$ .
Find the excluded values of $r (x) = \frac{x + 4}{x ^{2} - 9}$ .

Practice polynomials in the Greek quiz

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