1.4 Polynomials
3 min read•june 24, 2024
Polynomials are mathematical expressions with variables and exponents. They're the building blocks of algebra, used to model real-world situations and solve complex problems. Understanding polynomials is crucial for grasping more advanced mathematical concepts.
In this section, we'll cover the basics of polynomials and how to perform operations with them. We'll learn about , leading coefficients, and different forms of polynomials. We'll also practice adding, subtracting, and multiplying polynomials, including those with multiple variables.
Polynomial Basics
Degree and leading coefficient
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- of a polynomial is the highest exponent of the variable in the polynomial
- has a degree of 4 since the highest exponent is 4
- is the of the term with the highest degree
- has a of 3, the coefficient of the term
Polynomial Forms and Characteristics
- of a polynomial arranges terms in descending order of degree ()
- describes how a behaves as x approaches positive or negative infinity
- Polynomials can be divided using , similar to regular long division
Polynomial Operations
Addition and subtraction of polynomials
- Add or subtract polynomials by combining (terms with the same variables and exponents)
- combines like terms and , and , and and
- Subtract polynomials by distributing the negative sign to each term in the second polynomial before combining like terms
- distributes the negative sign to , , and , then combines like terms
Multiplication of polynomials
- Multiply polynomials using the distributive property, multiplying each term in the first polynomial by each term in the second polynomial
- multiplies by and , and by and , then combines like terms
- is a mnemonic for multiplying two binomials: First, Outer, Inner, Last
- Multiply the first terms, the outer terms, the inner terms, and the last terms, then combine like terms
- multiplies by , by , by , and by , then combines like terms
Operations with multiple variables
- Apply the same rules for addition, subtraction, and multiplication when working with polynomials containing multiple variables
- combines like terms and , and , and and
- Use the product rule for exponents when multiplying terms with the same variable in polynomials with multiple variables
- multiplies the coefficients and adds the exponents of like variables
Simplification of complex polynomials
- Simplify complex polynomial expressions by breaking them down into smaller parts and applying the appropriate operations (addition, subtraction, multiplication)
- can be simplified using these steps:
- multiplies the binomials
- multiplies the binomials
- subtracts the resulting polynomials by distributing the negative sign and combining like terms
- can be simplified using these steps:
Key Terms to Review (40)
Binomial: A binomial is a polynomial with exactly two terms. It is a mathematical expression that consists of two monomials, which are variables or constants combined by addition or subtraction.
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.
Completing the square: Completing the square is a method to solve quadratic equations by converting them into a perfect square trinomial. This facilitates easier solving and helps in deriving the quadratic formula.
Completing the Square: Completing the square is a technique used to solve quadratic equations and transform quadratic functions into a more useful form. It involves rearranging a quadratic expression into a perfect square plus or minus a constant, allowing for easier analysis and manipulation of the equation or function.
Constant Term: The constant term is a numerical value in a polynomial or equation that does not depend on any variable. It is the term that remains unchanged regardless of the values assigned to the variables in the expression.
Cubic Equation: A cubic equation is a polynomial equation of degree three, where the highest exponent of the variable is three. These equations are used to model a variety of real-world phenomena and are an important part of the study of polynomials.
Cubic Polynomial: A cubic polynomial is a polynomial of degree three, meaning it contains a term with the variable raised to the power of three. Cubic polynomials are an important class of functions in algebra and have unique properties that distinguish them from linear and quadratic polynomials.
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.
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.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
FOIL: FOIL is a method used for multiplying two binomials. It stands for First, Outer, Inner, Last, which are the steps involved in the multiplication process.
FOIL Method: The FOIL method is a systematic approach used to multiply two binomials, or expressions with two terms. The acronym FOIL stands for the order in which the terms are multiplied: First, Outer, Inner, Last.
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.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Leading term: The leading term of a polynomial is the term with the highest power of the variable. It determines the end behavior of the polynomial function.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, as they represent the same type of quantity.
Monomial: A monomial is a polynomial with only one term, which can be a constant, a variable, or a product of constants and variables raised to non-negative integer powers.
Monomial: A monomial is a single algebraic term that consists of a numerical coefficient and one or more variables raised to non-negative integer powers. It is the fundamental building block of polynomials, which are expressions formed by the sum of monomials.
Polynomial Division: Polynomial division is the process of dividing one polynomial by another to find the quotient and remainder. It is a fundamental operation in algebra that allows for the factorization and simplification of polynomial expressions.
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.
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.
Quadratic equation: A quadratic equation is a second-degree polynomial equation in a single variable, typically written as $ax^2 + bx + c = 0$, where $a \neq 0$. The solutions to the quadratic equation are known as the roots of the equation.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in mathematics and have applications in various fields, including physics, engineering, and economics.
Quadratic Polynomial: A quadratic polynomial is a polynomial expression with the highest exponent being 2. It is a mathematical function that can be represented in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic polynomials are widely used in various mathematical and scientific applications, including optimization problems, physics, and engineering.
Root: In the context of polynomials, a root is a value of the variable that makes the polynomial equation equal to zero. Roots represent the x-intercepts or solutions to the polynomial function, and are a fundamental concept in understanding polynomial behavior and properties.
Standard form: Standard form of a linear equation in one variable is written as $Ax + B = 0$, where $A$ and $B$ are constants and $x$ is the variable. The coefficient $A$ should not be zero.
Standard Form: Standard form is a way of expressing mathematical equations or functions in a specific, organized format. It provides a consistent structure that allows for easier manipulation, comparison, and analysis of these mathematical representations across various topics in algebra and beyond.
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 $x - c$.
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.
Term of a polynomial: A term of a polynomial is an expression consisting of a coefficient and one or more variables raised to non-negative integer exponents. Each term in a polynomial is separated by addition or subtraction operators.
Trinomial: A trinomial is a polynomial with exactly three terms, usually written in the form $ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. It is a specific type of polynomial that often appears in quadratic equations.
Turning point: A turning point is a point on the graph of a polynomial function where the graph changes direction from increasing to decreasing or vice versa. It occurs at local maxima or minima.
Turning Point: A turning point is a critical moment or event that marks a significant change or shift in direction, often serving as a pivotal point that can alter the course of something. This term is particularly relevant in the context of analyzing the behavior and characteristics of various mathematical functions, including polynomials, power functions, and parabolas.
X-intercept: The x-intercept is the point where a graph crosses the x-axis, where the y-coordinate is zero. It represents the solution(s) to an equation when $y = 0$.
X-Intercept: The x-intercept of a graph is the point where the graph of a function or equation intersects the x-axis, indicating the value of x when the function's output or the equation's value is zero. The x-intercept is a crucial concept in understanding the behavior and properties of various mathematical functions and equations.
Zero: In mathematics, the term 'zero' refers to the numerical value that represents the absence of quantity or magnitude. It is a fundamental concept that serves as the starting point for numerical systems and various mathematical operations.