Question

When a real zero $x=a$ of a polynomial function $f$ is of even multiplicity, the graph of $f$ _ the $x$ -axis at $x=a,$ and when it is of odd multiplicity, the graph of $f$ _ the $x$ -axis at $x=a$.

Name: Problem 5, Chapter 2: Precalculus
Uploaded: 2021-02-08T21:53:50-08:00
Duration: 3 min 12 s
Channel: Joshua Roe
Description: When a real zero x=a of a polynomial function f is of even multiplicity, the graph of f the x -axis at x=a, and when it is of odd multiplicity, the graph of f the x -axis at x=a.

   When a real zero $x=a$ of a polynomial function $f$ is of even multiplicity, the graph of $f$ _______ the $x$ -axis at $x=a,$ and when it is of odd multiplicity, the graph of $f$ _______ the $x$ -axis at $x=a$.

Precalculus

Ron Larson 10th Edition

Chapter 2, Problem 5

Instant Answer

Solved by Expert Joshua Roe

02/08/2021

Step 1

For the given polynomial function $f(x) = 4x^4(x-1)(x+1)$, we can see that the real zeros are $x=0$, $x=1$, and $x=-1$. Show more…

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When a real zero $x=a$ of a polynomial function $f$ is of even multiplicity, the graph of $f$ _______ the $x$ -axis at $x=a,$ and when it is of odd multiplicity, the graph of $f$ _______ the $x$ -axis at $x=a$.

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Key Concepts

Graph Behavior at Zeros

The behavior of a polynomial graph near its zeros is influenced by the multiplicity of those zeros. Even multiplicities cause the graph to merely touch the axis without crossing, often leading to a turning point, while odd multiplicities result in the graph crossing the axis, indicating a sign change in the function.

Multiplicity of a Zero

The multiplicity of a zero refers to how many times that particular zero occurs in the polynomial equation. It determines the nature of the graph at that point. If the multiplicity is even, the graph touches the x-axis and turns around; if it is odd, the graph crosses the x-axis.

Zeros of a Polynomial

A zero of a polynomial is a solution to the equation where the polynomial equals zero. These values indicate where the graph of the function intersects the x-axis. Understanding zeros helps in analyzing the behavior of the function and in constructing its graph.

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