Enrol to start learning
Reading is open to everyone. Enrolling is free, and it is what unlocks the audio lessons, practice tests and progress tracking.
1. What is a Polynomial?
Interactive Audio Lesson
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Today, we are discussing polynomials! A polynomial is an expression that includes variables and coefficients, alongside operations like addition and multiplication. Does anyone know what makes up a polynomial?
It has variables and coefficients, right?
Exactly! The coefficients are real numbers, and the variables can take different values. Can someone give an example of a polynomial?
What about 4x³ - 2x² + 7x - 5?
Great example! This polynomial has a degree of 3, which is the highest power of the variable. Remember, the degree is crucial when classifying polynomials.
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Now that we understand what a polynomial is, let's dive into the types. Polynomials can be classified based on their degree or the number of terms. For instance, a polynomial of degree 0 is called a constant polynomial. Can anyone name one?
P(x) = 5 is a constant polynomial!
Correct! And what about a linear polynomial?
P(x) = 3x + 2 would be a linear polynomial since it has a degree of 1.
Exactly! Remember to classify the polynomials correctly based on their degrees and number of terms, such as monomial, binomial, and trinomial.
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free accountLet's talk about the degree of a polynomial more in-depth. The degree refers to the highest power of the variable where the coefficient is non-zero. Can anyone give me an example of finding the degree?
If I take P(x) = 7x⁴ - x² + 3, the degree would be 4.
Well done! Understanding the degree is vital because it helps when performing operations on polynomials. What’s the degree in our previous example of 4x³ - 2x² + 7x - 5?
That would also be 3!
Perfect! You all are grasping these concepts. Remember, the degree influences how polynomials behave in graphs.
Reference YouTube Videos
Audio Book
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountA polynomial in one variable x is an expression of the form:
𝑃(𝑥) = 𝑎𝑛𝑥^𝑛 + 𝑎𝑛−1𝑥^{n−1} + ⋯ + 𝑎1𝑥 + 𝑎0
where:
• 𝑎0, 𝑎1, ..., 𝑎𝑛 are real numbers (coefficients)
• 𝑥 is a variable
• 𝑛 is a non-negative integer (degree of the polynomial)
Detailed Explanation
A polynomial is a type of mathematical expression that involves numbers and variables combined using addition, subtraction, multiplication, and non-negative integer exponents. In the expression given, 'P(x)' is the polynomial, and 'x' is the variable we can vary. The coefficients (like 'a0', 'a1', etc.) are real numbers that define how much each term contributes to the polynomial. The highest degree 'n' tells us the polynomial's complexity.
Examples & Analogies
Think of a polynomial like a recipe that tells you how many cups of different ingredients (coefficients) to mix together (terms). Each ingredient adds its own flavor depending on its amount, just like each term in a polynomial influences the overall value based on how big the variable is.Operations allowed in a polynomial include only addition, subtraction, and multiplication—emphasizing that division by a variable or negative exponents disqualify an expression from being a polynomial.
Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Polynomial: An algebraic expression involving variables raised to non-negative integer powers and coefficients.
Degree: The highest exponent of a variable in a polynomial expression.
Coefficient: A numerical factor in a polynomial expression.
Types of Polynomials: Includes constant, linear, quadratic, and cubic based on their degree.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Stories
Memory Tools
Flash Cards
Glossary
Polynomial
A mathematical expression consisting of variables, coefficients, and operations of addition, subtraction, and multiplication with non-negative integer exponents.
Coefficient
A real number multiplying a variable in a polynomial.
Degree
The highest power of the variable in a polynomial expression.
Constant Polynomial
A polynomial of degree 0, such as P(x) = 5.
Linear Polynomial
A polynomial of degree 1, such as P(x) = 3x + 2.
Quadratic Polynomial
A polynomial of degree 2, such as P(x) = x² - 4x + 4.
Cubic Polynomial
A polynomial of degree 3, such as P(x) = x³ - 3x² + x - 2.
Monomial
A polynomial with one term.
Binomial
A polynomial with two terms.
Trinomial
A polynomial with three terms.