Fiveable
Fiveable
College Algebra
Unit 5 Overview: Polynomial and Rational Functions
5.1 Quadratic Functions
5.2 Power Functions and Polynomial Functions
5.3 Graphs of Polynomial Functions
5.4 Dividing Polynomials
5.5 Zeros of Polynomial Functions
5.6 Rational Functions
5.7 Inverses and Radical Functions
5.8 Modeling Using Variation

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5.5 Zeros of Polynomial Functions

Last Updated on June 24, 2024

Polynomial division and zeros are crucial concepts in algebra. They help us understand how polynomials behave and break down into simpler parts. These tools are essential for solving equations and analyzing functions in real-world scenarios.

The remainder theorem and factor theorem connect polynomial division to finding zeros. By using these theorems along with techniques like the rational zero theorem, we can solve complex polynomial equations and gain insights into their properties.

Polynomial Division and the Remainder Theorem

Remainder theorem for polynomial evaluation

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  • States remainder when polynomial P(x)P(x) divided by x−cx - c equals P(c)P(c)
    • Find remainder when P(x)P(x) divided by x−cx - c by evaluating P(c)P(c)
  • Polynomial long division divides polynomial by linear factor
    • Results in polynomial quotient and remainder
    • Degree of remainder always less than degree of divisor

Zeros and Factors of Polynomial Functions

Factor theorem for polynomial zeros

  • States x−cx - c is factor of polynomial P(x)P(x) if and only if P(c)=0P(c) = 0
    • If P(c)=0P(c) = 0, then cc is zero (root) of polynomial function
  • Find factors of polynomial by setting equal to zero and solving for xx
    • Each solution corresponds to linear factor of form x−cx - c
  • Complex zeros may occur in conjugate pairs for polynomials with real coefficients

Rational zero theorem applications

  • States if polynomial has integer coefficients, rational zero must be ±pq\pm \frac{p}{q}
    • pp is factor of constant term
    • qq is factor of leading coefficient
  • Find potential rational zeros by listing possible ±pq\pm \frac{p}{q} combinations and testing each using factor theorem

Methods for finding polynomial zeros

  • Factoring: If polynomial can be factored, set each factor to zero and solve for xx
  • Quadratic Formula: For quadratic polynomials, use quadratic formula to find zeros
  • Rational Zero Theorem: Use to find potential rational zeros, then test each using factor theorem
  • Polynomial Long Division: If zero is known, use to reduce polynomial's degree and find additional zeros

Linear factorization theorem usage

  • States polynomial P(x)P(x) with real coefficients can be written as product of linear factors:
    • P(x)=a(x−r1)(x−r2)...(x−rn)P(x) = a(x - r_1)(x - r_2)...(x - r_n), where aa is leading coefficient and r1,r2,...,rnr_1, r_2, ..., r_n are zeros of P(x)P(x)
  • Construct polynomial with specified zeros by multiplying linear factors of form x−rix - r_i for each zero, then expanding product

Analyzing Polynomial Functions

Characteristics of Polynomial Functions

  • Polynomial function is an expression consisting of variables and coefficients using only addition, subtraction, and multiplication operations
  • Degree of a polynomial is the highest power of the variable in the function
  • End behavior describes how the function values change as x approaches positive or negative infinity
  • The number of zeros (real and complex) equals the degree of the polynomial

Descartes' rule of signs

  • States:
    1. Number of positive real zeros either equals number of sign changes between consecutive nonzero coefficients or is less by even number
    2. Number of negative real zeros either equals number of sign changes between consecutive nonzero coefficients of P(−x)P(-x) or is less by even number
  • Estimate number of positive real zeros by counting sign changes between consecutive nonzero coefficients
  • Estimate number of negative real zeros by replacing xx with −x-x in polynomial and counting sign changes between consecutive nonzero coefficients

Real-world polynomial problem solving

  1. Identify key components of problem and assign variables to unknown quantities
  2. Create polynomial equation that models problem situation
  3. Use appropriate methods to solve polynomial equation (factoring, rational zero theorem, polynomial long division)
  4. Interpret solution(s) in context of original problem, considering limitations or constraints

Key Terms to Review (33)

Coefficient: A coefficient is a numerical or constant factor that multiplies a variable in a term of an algebraic expression or equation. For example, in the term $5x^2$, 5 is the coefficient.
Complex Conjugate Theorem: The Complex Conjugate Theorem states that if a polynomial has real coefficients, then any non-real complex zeros must occur in conjugate pairs. If $a + bi$ is a zero, then $a - bi$ must also be a zero.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Cubic functions: A cubic function is a polynomial function of degree three, typically expressed in the form $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants and $a \neq 0$. These functions can have up to three real roots and exhibit distinct characteristics such as points of inflection.
Descartes’ Rule of Signs: Descartes’ Rule of Signs is a method used to determine the number of positive and negative real zeros in a polynomial function. It analyzes the sign changes in the coefficients of the polynomial.
Multiplicity: Multiplicity of a root in a polynomial function is the number of times that root occurs. It affects the shape and behavior of the graph at the corresponding $x$-intercept.
Polynomial function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can be expressed in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $a_i$ are constants and $n$ is a non-negative integer.
Roots: Roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as solutions or zeros of the equation.
Rational Zero Theorem: The Rational Zero Theorem states that any rational root of a polynomial equation with integer coefficients is a fraction $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
Synthetic division: Synthetic division is a simplified method of dividing polynomials where only the coefficients are used. It is particularly useful for dividing by linear factors of the form $x - c$.
Zeros: Zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation $P(x) = 0$.
Complex Conjugates: Complex conjugates are a pair of complex numbers that have the same real part, but their imaginary parts have opposite signs. This relationship between complex numbers is important in the study of polynomial functions and partial fractions.
Complex Zeros: Complex zeros are the roots of a polynomial function that are complex numbers, meaning they have both a real and an imaginary component. These zeros are important in understanding the behavior and properties of polynomial functions.
Cubic Function: A cubic function is a polynomial function of degree three, where the highest exponent of the variable is three. Cubic functions have a distinctive S-shaped curve and can exhibit a variety of behaviors, including having one, two, or three real zeros, depending on the coefficients of the function.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
End Behavior: The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
Descartes' Rule of Signs: Descartes' Rule of Signs is a mathematical principle that provides information about the number of positive and negative real roots of a polynomial equation. It helps determine the possible number of sign changes in the coefficients of a polynomial, which corresponds to the number of positive and negative real roots.
F(x) = 0: The equation f(x) = 0 represents the points where a function intersects the x-axis, also known as the zeros or roots of the function. This equation is a fundamental concept in the study of polynomial functions and their properties.
Factor Theorem: The Factor Theorem is a fundamental principle in polynomial algebra that establishes a relationship between the factors of a polynomial and its zeros. It provides a way to determine whether a given expression is a factor of a polynomial and to find the roots or zeros of a polynomial function.
Factors: Factors are the elements or components that contribute to the formation, development, or determination of a particular outcome or phenomenon. In the context of polynomial functions, factors are the values or expressions that, when multiplied together, result in the original polynomial equation.
Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex number coefficients has at least one complex number solution. It is a foundational result in algebra that connects the properties of polynomials to the nature of complex numbers.
Linear Factorization Theorem: The Linear Factorization Theorem states that any polynomial function of degree n can be expressed as a product of n linear factors. This theorem provides a fundamental way to understand the relationship between the roots or zeros of a polynomial function and its factored form.
Multiplicity: Multiplicity refers to the number of times a particular value, known as a zero or root, occurs in the factorization of a polynomial function. It is an important concept in understanding the behavior of polynomial graphs and the nature of their zeros.
Polynomial Function: A polynomial function is an algebraic function that can be expressed as the sum of a finite number of non-negative integer powers of a variable, with coefficients. Polynomial functions are a fundamental concept in algebra and are closely related to topics such as power functions, polynomial division, and the zeros of polynomial functions.
Polynomial Long Division: Polynomial long division is a method used to divide one polynomial by another polynomial. It involves a step-by-step process of dividing the terms of the dividend by the terms of the divisor, similar to the long division algorithm used for dividing integers.
Quartic Function: A quartic function is a polynomial function of degree four, meaning it has the general form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are real numbers. Quartic functions are a specific type of polynomial function that are important in the study of graphs of polynomial functions and the determination of their zeros.
Remainder Theorem: The Remainder Theorem is a fundamental principle in polynomial division that states the relationship between the division of a polynomial by a linear expression and the value of the polynomial when the variable is set to a specific value. It provides a way to determine the remainder of a polynomial division without actually performing the long division process.
Roots: In mathematics, the term 'roots' refers to the solutions or values of a polynomial equation that make the equation equal to zero. Roots are an essential concept in various topics related to polynomial functions and equations, including quadratic equations, power functions, and the graphs of polynomial functions.
Rational Root Theorem: The Rational Root Theorem is a fundamental principle in the study of polynomial functions that provides a way to determine the possible rational roots of a polynomial equation. It helps simplify the process of finding the roots or zeros of a polynomial by narrowing down the potential solutions.
Synthetic Division: Synthetic division is a shortcut method used to divide a polynomial by a linear expression of the form $(x - a)$. It allows for the efficient computation of polynomial division, providing a streamlined approach to determining the quotient and remainder of the division process.
Turning Points: Turning points are the critical points on the graph of a polynomial function where the direction of the curve changes. These points mark the transition between increasing and decreasing behavior, and are essential in understanding the overall shape and behavior of polynomial functions.
X-Intercepts: The x-intercepts of a graph are the points where the graph crosses the x-axis, or where the function's y-value is equal to zero. They represent the values of x for which the function has a solution of y = 0, and they provide important information about the behavior and characteristics of the function.
Zeros: Zeros, also known as roots, are the values of the independent variable that make a function equal to zero. They are the points where the graph of a function intersects the x-axis, representing the solutions to the equation f(x) = 0.