Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Graphs of Polynomial Functions: Finding Zeros, Multiplicities, and Turning Points - Prof. , Study notes of Mathematics

A comprehensive guide to understanding and analyzing graphs of polynomial functions. It covers key concepts such as finding zeros using factoring, identifying multiplicities of zeros, interpreting turning points, and applying the intermediate value theorem. Well-structured, includes illustrative examples, and provides practical exercises for reinforcement.

Typology: Study notes

2023/2024

Uploaded on 12/11/2024

miabella-correa
miabella-correa 🇺🇸

4 documents

1 / 15

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
3.4 GRAPHS OF POLYNOMIAL FUNCTIONS
Using Factoring to Find Zeros of Polynomial Functions
Given a polynomial function f, find the x-intercepts by factoring.
1. Set f (x) = 0.
2. If the polynomial function is not given in factored form:
a. Factor out any common monomial factors.
b. Factor any factorable binomials or trinomials.
3. Set each factor equal to zero and solve to find the x-intercepts
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

Partial preview of the text

Download Graphs of Polynomial Functions: Finding Zeros, Multiplicities, and Turning Points - Prof. and more Study notes Mathematics in PDF only on Docsity!

3.4 GRAPHS OF POLYNOMIAL FUNCTIONS

Using Factoring to Find Zeros of Polynomial Functions Given a polynomial function f, find the x-intercepts by factoring.

  1. Set f (x) = 0.
  2. If the polynomial function is not given in factored form: a. Factor out any common monomial factors. b. Factor any factorable binomials or trinomials.
  3. Set each factor equal to zero and solve to find the x-intercepts

EX. Find the x-intercepts of f (x) = x 6 − 3x 4

  • 2x 2

EX. Find the x-intercepts of h(x) = x 3

  • 4x 2
  • x − 6. This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts.

Try It # Find the y- and x-intercepts of the function f (x) = x^4 − 19x^2 + 30x.

If a polynomial contains a factor of the form (x − h) p , the behavior near the x-intercept h is determined by the power p. We say that x = h is a zero of multiplicity p. The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. The graph will cross the x-axis at zeros with odd multiplicities. The sum of the multiplicities is the degree of the polynomial function.

EX. Use the graph of the function of degree 6 in Figure 9 to identify the zeros of the function and their possible multiplicities.  The first zero occurs at x = −3. The graph touches the x-axis, so the multiplicity of the zero must be even. The zero of −3 has multiplicity 2.  The next zero occurs at x = −1. The graph looks almost linear at this point. This is a single zero of multiplicity 1.  The last zero occurs at x = 4. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.

Intermediate Value Theorem Let f be a polynomial function. The Intermediate Value Theorem states that if f (a) and f (b) have opposite signs, then there exists at least one value c between a and b for which f (c) = 0.

EX. Show that the function f (x) = x 3 − 5x 2

  • 3x + 6 has at least two real zeros between x = 1 and x = 4. We see that one zero occurs at x = 2. Also, since f (3) is negative and f (4) is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4. We have shown that there are at least two real zeros between x = 1 and x = 4.

Try It # Given the graph below, write a formula for the function shown.