Glossary

3.6 Zeros of Polynomial Functions

4 min read•june 24, 2024

Polynomial division and the are powerful tools for working with polynomials. They simplify calculations and help us find zeros quickly. These techniques are crucial for understanding polynomial behavior and solving related problems.

Zeros of polynomial functions are key to factoring and graphing. By using the and Theorem, we can find real and complex zeros, which reveal important information about a polynomial's structure and behavior.

Polynomial Division and the Remainder Theorem

Application of Remainder Theorem

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  • Remainder Theorem efficiently evaluates polynomials when divided by a (x−c)(x - c)
    • Calculates the remainder by evaluating the polynomial at cc, P(c)P(c)
    • Avoids the need for long division, simplifying the process
  • Example: For P(x)=x3−4x2+5x−2P(x) = x^3 - 4x^2 + 5x - 2, the remainder when divided by x−3x - 3 is P(3)=33−4(3)2+5(3)−2=27−36+15−2=4P(3) = 3^3 - 4(3)^2 + 5(3) - 2 = 27 - 36 + 15 - 2 = 4
  • can be used for more complex divisions or when the full quotient is needed

Zeros of Polynomial Functions

Factor Theorem for polynomial zeros

  • Factor Theorem determines if a value cc is a () of a polynomial P(x)P(x)
    • If P(c)=0P(c) = 0, then cc is a zero and (x−c)(x - c) is a factor of P(x)P(x)
    • Conversely, if (x−c)(x - c) is a factor, then cc is a zero of P(x)P(x)
  • Example: For P(x)=x3−3x2−4x+12P(x) = x^3 - 3x^2 - 4x + 12, P(2)=23−3(2)2−4(2)+12=8−12−8+12=0P(2) = 2^3 - 3(2)^2 - 4(2) + 12 = 8 - 12 - 8 + 12 = 0, so 2 is a zero and (x−2)(x - 2) is a factor

Rational Zero Theorem applications

  • Rational Zero Theorem lists potential rational zeros of a polynomial with integer coefficients
    • Possible zeros are in the form ±pq\pm \frac{p}{q}, where pp is a factor of the constant term and qq is a factor of the leading coefficient
    • Test potential zeros using the Factor Theorem to find actual zeros
  • Example: For P(x)=3x3−5x2−11x+15P(x) = 3x^3 - 5x^2 - 11x + 15, potential rational zeros are ±1,±3,±5,±15,±13,±53\pm 1, \pm 3, \pm 5, \pm 15, \pm \frac{1}{3}, \pm \frac{5}{3}

Classification of polynomial zeros

  • Polynomial zeros can be real (rational or irrational) or complex
    • Real zeros are found using the Rational Zero Theorem and Factor Theorem
    • Complex zeros occur in conjugate pairs (a+bi)(a + bi) and (a−bi)(a - bi)
  • Zeros are classified by :
    • Single zeros have multiplicity 1 and the graph crosses the x-axis
    • Repeated zeros have multiplicity greater than 1 and the graph touches the x-axis
  • Example: P(x)=(x−2)(x−3)2(x−(1+i))(x−(1−i))P(x) = (x - 2)(x - 3)^2(x - (1 + i))(x - (1 - i)) has a at 2, a at 3, and complex zeros at 1±i1 \pm i
  • The can be used to find zeros of quadratic polynomials, including complex zeros

Linear Factorization Theorem usage

  • expresses a polynomial as a product of linear and irreducible quadratic factors
    • Linear factors (x−c)(x - c) correspond to real zeros
    • Irreducible quadratic factors (x−(a+bi))(x−(a−bi))(x - (a + bi))(x - (a - bi)) correspond to complex zeros
  • Constructing a polynomial with specified zeros involves multiplying the corresponding factors
    • Include repeated factors for zeros with multiplicity greater than 1
  • Example: A polynomial with zeros -1, 2 (with multiplicity 3), and 3−2i3 - 2i is P(x)=(x+1)(x−2)3(x−(3−2i))(x−(3+2i))P(x) = (x + 1)(x - 2)^3(x - (3 - 2i))(x - (3 + 2i))

Analyzing Polynomial Functions

Descartes' Rule of Signs

  • ###'_Rule_of_Signs_0### estimates the number of positive and negative real zeros
    • The number of positive real zeros is either the number of sign changes between consecutive terms or less by an even number
    • For negative real zeros, consider sign changes in P(−x)P(-x)
  • Example: P(x)=2x4−3x3−12x2+5x+8P(x) = 2x^4 - 3x^3 - 12x^2 + 5x + 8 has 2 sign changes, so there are either 2 or 0 positive real zeros. P(−x)P(-x) has 3 sign changes, so there are either 3 or 1 negative real zeros.

Polynomial modeling in real-world problems

  1. Identify the key variables and relationships in the problem
  2. Construct a polynomial equation representing the situation
  3. Solve the polynomial equation using appropriate methods (factoring, Rational Zero Theorem, etc.)
  4. Interpret the solution in the context of the problem, considering limitations and constraints
  • Example: A rectangular prism has a square base and a volume of 250 cubic centimeters. If the height is 3 centimeters more than the side length of the base, find the dimensions.
    • Let xx be the side length of the base, then the height is x+3x + 3
    • The volume is given by x2(x+3)=250x^2(x + 3) = 250
    • Solve x3+3x2−250=0x^3 + 3x^2 - 250 = 0 to find the dimensions (length, width, and height) of the prism

Graphing and Analyzing Polynomial Functions

  • helps visualize their behavior and identify key features
    • x-intercepts correspond to real zeros of the polynomial
    • y-intercept is the constant term of the polynomial
    • of the graph is determined by the degree and leading coefficient of the polynomial
  • Analyzing the graph can provide insights into the number and types of zeros, including

Key Terms to Review (27)

Complex Conjugates: Complex conjugates are a pair of complex numbers that have the same real part but opposite imaginary parts. They are fundamental in understanding the behavior of polynomial functions and their roots.
Complex Numbers: Complex numbers are a mathematical concept that extend the real number system to include both real and imaginary components. They are represented in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, with $i$ representing the imaginary unit (the square root of -1).
Complex Zero: A complex zero is a root or solution of a polynomial equation where the root is a complex number, meaning it has both a real and an imaginary component. Complex zeros are an important concept in the study of polynomial functions and their behavior.
Cubic Function: A cubic function is a polynomial function of degree three, meaning it contains a variable raised to the third power. These functions have a distinctive S-shaped curve and can be used to model a wide range of real-world phenomena, from population growth to the trajectory of projectiles.
Descartes: Descartes, also known as René Descartes, was a renowned French philosopher, mathematician, and scientist who lived in the 17th century. His philosophical ideas and mathematical contributions have had a profound impact on various fields, including the understanding of polynomial functions and parametric equations.
Descartes' Rule of Signs: Descartes' Rule of Signs is a mathematical principle that provides a way to determine the possible number of positive and negative real roots of a polynomial function. It is a useful tool for analyzing the behavior and characteristics of polynomial functions.
End Behavior: The end behavior of a function refers to the behavior or pattern of the function as it approaches positive or negative infinity. It describes the long-term trends and asymptotic properties of the function, providing insights into how the function will continue to behave outside of the given domain.
Factor Theorem: The Factor Theorem is a fundamental principle in polynomial algebra that establishes a connection between the zeros of a polynomial function and the factors of that polynomial. It provides a systematic way to determine whether a particular value is a root or zero of a 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 root. This theorem establishes a fundamental connection between the algebraic properties of polynomial equations and the geometric properties of the complex number plane.
Graphing Polynomials: Graphing polynomials is the process of visually representing the behavior and characteristics of polynomial functions on a coordinate plane. This involves determining the key features of the polynomial, such as its degree, zeros, and end behavior, and then using that information to sketch an accurate graph of the function.
Irrational Zero: An irrational zero of a polynomial function is a root or solution of the equation that cannot be expressed as a ratio of two integers. These zeros are represented by irrational numbers, which have decimal expansions that never terminate or repeat. Irrational zeros are an important consideration when analyzing the behavior and properties of polynomial functions.
Irreducible Quadratic Factor: An irreducible quadratic factor is a polynomial expression of degree two that cannot be factored any further into smaller polynomial expressions. It represents a quadratic equation that has no real-world solutions that can be expressed as the product of two linear factors.
Linear Factor: A linear factor is a factor of a polynomial expression that is a linear expression, meaning it is a polynomial of degree 1. These factors are important in the context of dividing polynomials and finding the zeros of polynomial functions.
Linear Factorization Theorem: The Linear Factorization Theorem states that any polynomial function can be expressed as a product of linear factors. This theorem provides a way to find the zeros of a polynomial function by identifying its linear factors.
Multiplicity: Multiplicity refers to the number of times a particular value occurs as a zero or root of a polynomial function. It describes the order or repetition of a zero or root within the function's graph or equation.
Polynomial Function: A polynomial function is a mathematical function that is defined by a polynomial, which is an expression consisting of variables and coefficients with non-negative integer exponents. Polynomial functions are a fundamental class of functions in mathematics and have numerous applications in various fields, including science, engineering, and economics.
Polynomial Long Division: Polynomial long division is a method used to divide one polynomial by another polynomial. It is a systematic process of dividing the dividend polynomial by the divisor polynomial, similar to the long division algorithm used for dividing integers. This technique is particularly useful for finding the zeros or roots of a polynomial function.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are equations that contain a variable raised to the second power. This formula provides a systematic way to find the solutions, or roots, of any quadratic equation, regardless of the coefficients involved.
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 important in the study of polynomial functions and their properties, particularly in the context of finding the zeros or roots of such functions.
Rational Root Theorem: The Rational Root Theorem is a fundamental concept in the study of polynomial functions, which states that if a polynomial function with integer coefficients has a rational root, then that root must be a factor of the constant term of the polynomial.
Rational Zero: A rational zero is a real number that can be expressed as the ratio of two integers, where the denominator is not zero. Rational zeros are important in the context of polynomial functions, as they represent the points where the function intersects the x-axis.
Remainder Theorem: The Remainder Theorem is a fundamental concept in polynomial arithmetic that allows us to determine the remainder when a polynomial is divided by a linear expression of the form $(x - a)$. It provides a way to simplify the division process and understand the relationship between the divisor, the quotient, and the remainder.
Repeated Zero: A repeated zero is a root or solution of a polynomial equation that occurs more than once. It is a value of the independent variable that causes the polynomial function to evaluate to zero, and this value appears multiple times in the factored form of the polynomial.
Root: A root is a value that, when substituted into a function, results in the function evaluating to zero. Roots are essential in understanding the behavior and characteristics of polynomial functions, as well as the graphs of trigonometric functions.
Single Zero: A single zero of a polynomial function is a real number that, when substituted into the function, results in the function evaluating to zero. This means that the graph of the polynomial function will intersect the x-axis at that point, indicating a point where the function changes from positive to negative or vice versa.
Synthetic Division: Synthetic division is a shorthand method for dividing a polynomial by a linear expression, typically in the form of (x - a). This technique allows for efficient division of polynomials without the need for the full long division process, making it a valuable tool in the study of polynomial functions and their properties.
Zero: A zero is a point on the x-axis where a function intersects or touches the axis, indicating that the function's value is equal to zero at that point. Zeros are important in the study of polynomial functions and trigonometric functions, as they provide insights into the behavior and properties of these functions.
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