Polynomial Functions Overview

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Questions and Answers

What is the degree of the polynomial function mentioned in Problem 1?

4 (correct)
5
2
3

Which polynomial function has a leading coefficient that is negative?

$f(x) = x^{2} + 2x^{2}-4x^{3}+x$
$f(x) = -x^{4} + 2x^{2}$ (correct)
$f(x) = x^{3} - 5x$
$f(x) = x^{5} - 3x^{2} + 2$

What are the zeros and their multiplicities for the given polynomial in Problem 1?

Zero: 0, Multiplicity: 4; Zero: 1, Multiplicity: 1
Zero: -2, Multiplicity: 1; Zero: 1, Multiplicity: 3 (correct)
Zero: -2, Multiplicity: 3; Zero: 1, Multiplicity: 1
Zero: -1, Multiplicity: 2; Zero: 2, Multiplicity: 1

Which of the following functions corresponds to Graph A?

<p>$f(x) = x^{3} - 5x$ (C)</p> Signup and view all the answers

What is the factored form of the polynomial function in Problem 1?

<p>(x + 2)(x - 1) (B)</p> Signup and view all the answers

Flashcards

Leading Coefficient Positive

The coefficient of the highest-degree term in a polynomial function is positive.

Polynomial Function Degree

The degree of the polynomial is the highest power of the variable in the function's expression.

Zero and Multiplicity

A zero of a polynomial function is a value of x for which the function equals zero; multiplicity is the number of times a zero appears as a factor.

Polynomial Function's Factors

A polynomial function can be broken down into simpler expressions, known as factors. Factors are terms present when the function is written in factored form. These factors help in finding the roots or zeros of the function.

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Even/Odd Functions

A function is even if f(-x) = f(x) for all x. It is odd if f(-x) = -f(x).

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Study Notes

Polynomial Functions

Even/Odd Functions: Determine if a function is even or odd by analyzing its graph or equation
Leading Coefficient: The leading coefficient determines whether the graph goes up or down as x approaches positive or negative infinity
Degree: The degree of a polynomial function (the highest power of x) determines the end behavior and number of turning points
Zeros (and Multiplicity): Zeros are the x-values where the function crosses the x-axis. The multiplicity of each zero tells how many times that zero is repeated
Factored Form: A factored form of a polynomial equation can be used to find its zeros, which are the values where the function equals zero.