Polynomial

Welcome to the second episode of Algebra and Number Theory! Building on our understanding of linear functions, we now dive into the broader world of polynomials. In this episode, you'll learn to define and identify polynomials, mastering key terminology like 'degree', 'coefficients', and 'terms'. We'll explore how to classify them by their degree and number of terms. The core of our session will focus on performing fundamental operations: adding, subtracting, and multiplying polynomials by combining like terms and using the distributive property. Finally, we'll connect these algebraic expressions to their role as functions, setting the stage for solving complex problems in future episodes. By the end, you'll have a solid foundation in handling these versatile mathematical expressions.

Check your understanding

These are the same multiple-choice questions you will see in the Quiz section after you listen to the episode. Use them here to preview or review the answers.

Which of the following expressions are considered polynomials?

  1. 3x^2 + 2x - 1
  2. 5x^½ + 4
  3. 7x^-1
  4. 4x^3 - x
  5. 10

What is the degree of the polynomial 4x^3 - 8x^5 + x^2 + 7?

  1. 3
  2. 8
  3. 5
  4. 4
  5. 7

How can the polynomial 9x^2 - 4 be classified?

  1. Monomial
  2. Binomial
  3. Trinomial
  4. Linear
  5. Quadratic
  6. Cubic

What is the result of adding the polynomials (3x^2 - 5x + 2) and (x^2 + 2x - 1)?

  1. 4x^2 - 3x + 1
  2. 3x^2 - 3x + 1
  3. 4x^4 - 3x^2 + 1
  4. 4x^2 - 7x + 3

Which expression is the correct result of multiplying (x - 4) by (x + 2)?

  1. x^2 - 8
  2. x^2 + 2x - 8
  3. x^2 - 2x - 8
  4. x^2 - 6x - 8

Suggested next

Related episodes that are a natural follow-on.

  • Generating function

    Welcome to the fascinating world of generating functions! In this episode, we'll discover how these remarkable algebraic objects act as 'clotheslines' for sequences of numbers. You'll learn how to encode an entire infinite sequence into a single, com… Welcome to the fascinating world of generating functions! In this episode, we'll discover how these remarkable algebraic objects act as 'clotheslines' for sequences of numbers. You'll learn how to encode an entire infinite sequence into a single, compact function. This powerful technique transforms complex counting problems from combinatorics into more familiar tasks of manipulating polynomials and power series. We will explore the fundamental concept by building a generating function for a simple coin-toss experiment and then uncover the closed-form expression for one of the most important series in mathematics. This episode provides the foundation for solving even more advanced problems in future lessons.

  • Series (mathematics)

    Welcome to Episode 6: An Introduction to Series. Building on our understanding of limits and sequences, this episode dives into the fascinating world of infinite sums. What happens when you try to add up an infinite number of things? Sometimes, the s… Welcome to Episode 6: An Introduction to Series. Building on our understanding of limits and sequences, this episode dives into the fascinating world of infinite sums. What happens when you try to add up an infinite number of things? Sometimes, the sum explodes to infinity, but remarkably, it sometimes settles on a specific, finite value. We'll explore this fundamental distinction between divergence and convergence, using the concept of partial sums as our guide. You'll learn about the geometric series, a powerful and common type of series, and discover the simple condition that determines its sum. Finally, we'll see how series form a bridge to advanced calculus, allowing us to represent complex functions and solve otherwise difficult problems.

  • Diophantine equation

    Welcome to the third episode of our course! This time, we journey back to ancient Alexandria to explore Diophantine equations, named after the Greek mathematician Diophantus. These are not just any algebraic equations; their challenge lies in the que… Welcome to the third episode of our course! This time, we journey back to ancient Alexandria to explore Diophantine equations, named after the Greek mathematician Diophantus. These are not just any algebraic equations; their challenge lies in the quest for integer-only solutions. We will learn to identify when a linear Diophantine equation has solutions by connecting back to concepts like the greatest common divisor and modular arithmetic. You will discover the elegant Euclidean algorithm for finding a specific solution and learn how to generate all infinitely many solutions from just one. This episode lays the groundwork for understanding more complex problems in number theory, including the famous Fermat's Last Theorem, which we will tackle in a future session. Get ready to solve some ancient mathematical puzzles!

  • Number theory

    Welcome to the fourth episode of our course on Algebra and Number Theory! In this session, we venture into the elegant world of number theory, a field the great mathematician Carl Friedrich Gauss called 'the queen of mathematics.' This episode lays t… Welcome to the fourth episode of our course on Algebra and Number Theory! In this session, we venture into the elegant world of number theory, a field the great mathematician Carl Friedrich Gauss called 'the queen of mathematics.' This episode lays the groundwork by exploring the fundamental properties of integers, the building blocks of mathematics. We will delve into the core concepts of divisibility, understand the crucial division algorithm, and define the greatest common divisor (GCD). You will also learn about Euclid’s algorithm, an ancient and highly efficient method for finding the GCD. This knowledge builds upon your understanding of equations and provides a crucial foundation for our future exploration of prime numbers, rings, and fields.

  • Complex analysis

    Welcome to the third episode of Applied Mathematics! This time, we journey beyond the familiar number line into the fascinating world of **Complex Analysis**. You'll discover how the introduction of a single new number—the imaginary unit 'i'—opens up… Welcome to the third episode of Applied Mathematics! This time, we journey beyond the familiar number line into the fascinating world of **Complex Analysis**. You'll discover how the introduction of a single new number—the imaginary unit 'i'—opens up a whole new dimension of mathematics. This episode explains the fundamentals of complex numbers, how to visualize them on the complex plane, and what it means to have functions of a complex variable. We will explore the surprisingly powerful properties of 'analytic' functions and the elegance of complex integration, including Cauchy's remarkable theorems. By the end, you'll understand why this seemingly abstract field is an indispensable tool for engineers and physicists, providing elegant solutions to real-world problems.

Often studied before

Episodes that tend to come earlier on similar paths.

  • Coordinate geometry

    Welcome to the world of Coordinate Geometry, also known as Analytic Geometry! This episode forges a powerful connection between the abstract shapes of Euclidean geometry and the concrete numbers of algebra. We'll introduce the Cartesian coordinate sy… Welcome to the world of Coordinate Geometry, also known as Analytic Geometry! This episode forges a powerful connection between the abstract shapes of Euclidean geometry and the concrete numbers of algebra. We'll introduce the Cartesian coordinate system, a brilliant invention that allows us to place any point, line, or shape onto a two-dimensional grid. You will learn how to use coordinates to calculate the distance between two points and find the exact middle of a line segment. We will then explore how to translate geometric figures like lines and circles into the language of algebraic equations, unlocking a new way to analyze and solve geometric problems. This episode provides the foundational tools for describing geometry with the precision of algebra.

  • Mathematical notation

    Welcome to the final episode of our Foundations of Mathematics course! In this capstone session, we explore Mathematical Notation, the universal language that underpins all the topics we've covered, from Arithmetic to Set Theory. You will learn how t… Welcome to the final episode of our Foundations of Mathematics course! In this capstone session, we explore Mathematical Notation, the universal language that underpins all the topics we've covered, from Arithmetic to Set Theory. You will learn how this symbolic system evolved from lengthy verbal descriptions to the elegant and precise script we use today. We'll discover how notation provides the conciseness, clarity, and power of abstraction necessary to build complex arguments and communicate them across the globe. This episode ties together our entire journey, revealing how symbols turn abstract ideas into tangible tools for thought and discovery, solidifying your understanding of the mathematical world.

  • Mathematical proof

    Welcome to the sixth episode of Foundations of Mathematics. In this episode, we explore the heart of mathematical reasoning: the proof. Unlike other sciences that rely on evidence and observation, mathematics demands absolute certainty, and proofs ar… Welcome to the sixth episode of Foundations of Mathematics. In this episode, we explore the heart of mathematical reasoning: the proof. Unlike other sciences that rely on evidence and observation, mathematics demands absolute certainty, and proofs are the tools we use to achieve it. You will learn what constitutes a mathematical proof, discover its fundamental building blocks like axioms and theorems, and explore common methods such as direct proof and proof by contradiction. This episode will illuminate why proofs are not just about verification but are essential for building the logical, unshakeable structure of mathematics and providing deep understanding.

  • Vector space

    Welcome to the seventh episode of Algebra and Number Theory! This episode introduces the fundamental concept of a Vector Space. Building on your knowledge of linear algebra and matrices, we will move beyond the familiar idea of vectors as arrows in s… Welcome to the seventh episode of Algebra and Number Theory! This episode introduces the fundamental concept of a Vector Space. Building on your knowledge of linear algebra and matrices, we will move beyond the familiar idea of vectors as arrows in space. We will explore the abstract definition of a vector space through a specific set of rules, or axioms, that govern vector addition and scalar multiplication. You will discover that many different mathematical objects, including polynomials and matrices, can be considered vectors in their own right. This episode provides a crucial abstract foundation that unifies many areas of mathematics and prepares you for more advanced algebraic structures.

  • Function (mathematics)

    Welcome to the ninth episode of Foundations of Mathematics! Building on our understanding of sets, numbers, and algebra, we now introduce one of the most central concepts in all of mathematics: the function. In this episode, you'll learn what a funct… Welcome to the ninth episode of Foundations of Mathematics! Building on our understanding of sets, numbers, and algebra, we now introduce one of the most central concepts in all of mathematics: the function. In this episode, you'll learn what a function is, viewing it as a special kind of relationship or a 'machine' that takes an input and produces a unique output. We will explore its essential components, including the domain, codomain, and range, and understand the strict rule that defines a function. This concept is crucial, forming the basis for calculus, analysis, and countless applications in science and engineering. Get ready to uncover the precise and powerful idea of a mathematical function.