Degree of Polynomials

Hello there! Let’s dive into understanding the degree of polynomials with some fun and easy-to-digest explanations, complete with emojis and practice questions! 🎓✨

What is a Polynomial? 🤔

From a “Layman’s perspective,” a polynomial is a mathematical expression that’s essentially a sum of several mathematical terms. Each of these “mathematical terms” can be a number, a variable, or a product of several variables.

However, a “mathematician’s perspective” gives us a more precise definition: a polynomial is an algebraic expression where the only arithmetic operations allowed are addition, subtraction, multiplication, and variables can only have “natural exponents”. “Natural exponents” mean non-negative integers (0, 1, 2, and so on). For instance, an expression like “t raised to half plus t” (t^(1/2) + t) would not qualify as a polynomial because t^(1/2) has a rational exponent, not a natural one.

The word “polynomial” itself comes from two words: “poly” (meaning “many”) and “nomen” (meaning “names” or “terms”). So, it literally means an expression with “many terms”. Each individual term in a polynomial is called a monomial.

Understanding the “Degree” of a Polynomial 📏

The “degree” is a crucial property that tells us a lot about a polynomial. Here’s how we determine it:

1. Degree of a Variable in a Term 🧑‍🏫 The exponent on the variable in a term is called the “degree of that variable in that term”.

   • Example: In the term 4x²y², the degree of x is 2, and the degree of y is 2.

2. Degree of a Term ➕ The “degree of that term” is the sum of the degrees of the variables in that term.

   • Example: For the term 4x²y², the degree of the term is 2 (degree of x) + 2 (degree of y) = 4.
   • Another Example: For a constant term like 1, you can think of it as 1 * x⁰ * y⁰. So, the degree of a constant term is 0 + 0 = 0.

3. Degree of the Polynomial 👑 The “degree of the polynomial” is the largest degree among all the terms with non-zero coefficients within that polynomial.

   • Example: Let’s consider the polynomial 3x³ + 4x²y² + 10y + 1.
     • Term 1: 3x³
       • Degree of x is 3.
       • Degree of the term is 3.
     • Term 2: 4x²y²
       • Degree of x is 2, degree of y is 2.
       • Degree of the term is 2 + 2 = 4.
     • Term 3: 10y
       • Degree of y is 1.
       • Degree of the term is 1.
     • Term 4: 1 (constant)
       • Degree of the term is 0. Comparing the degrees of all terms (3, 4, 1, 0), the largest degree is 4. Therefore, the degree of the polynomial 3x³ + 4x²y² + 10y + 1 is 4.

Special Case: The Zero Polynomial 🚫

The degree of the zero polynomial is undefined. This is because the zero polynomial can be written with any power of x (e.g., 0 + 0x + 0x² + ...), and to define a degree, a non-zero coefficient is needed.

Classification of Polynomials by Degree 🏷️

Polynomials are often classified based on their degree:

• Degree 0: Constant polynomial (e.g., 5, π). Note: A constant polynomial can never be equal to 0 if its degree is defined as 0.
• Degree 1: Linear polynomial (e.g., 2x + 4, x + y). The crucial fact here is that the highest degree of any term is 1.
• Degree 2: Quadratic polynomial (e.g., 2x² + 4, ax² + bx + c). These functions represent parabolas when graphed.
• Degree 3: Cubic polynomial (e.g., 2x³ + 4). The highest monomial term will have a degree of 3.
• Degree 4: Quartic polynomial.
• Degree 5: Quintic or Quantic polynomial.
• In general, for degrees higher than 5, they are typically just referred to as “polynomials” of their respective degrees, though specific names exist for some.

This classification provides more specific information about the polynomial.

Practice Questions! 📝

Question 1: Identify if the following expression is a polynomial and, if so, determine its degree: f(x) = 5x⁴ - 2x + 7 + 9x²y³

Question 2: What is the degree of the polynomial represented by a straight line?

Question 3: Explain why the expression x + x^(1/2) is not considered a polynomial.

Solutions to Practice Questions ✅

Solution 1: Let’s break down f(x) = 5x⁴ - 2x + 7 + 9x²y³ term by term:

• Term 1: 5x⁴
  • Exponent on x is 4.
  • Degree of the term is 4.
• Term 2: -2x
  • Exponent on x is 1.
  • Degree of the term is 1.
• Term 3: 7 (constant)
  • Degree of the term is 0.
• Term 4: 9x²y³
  • Exponent on x is 2.
  • Exponent on y is 3.
  • Degree of the term is 2 + 3 = 5.

Since all exponents are natural numbers (non-negative integers), this is indeed a polynomial function [4.1 Definition of polynomial function:, 307].

The degrees of the terms are 4, 1, 0, and 5. The largest degree among these is 5. Therefore, f(x) is a polynomial, and its degree is 5. 🎉

Solution 2: A straight line can be represented by a linear function, which is a polynomial of degree 1. For example, y = mx + c where m is not zero. The highest exponent of the variable x is 1. Therefore, the degree of the polynomial represented by a straight line is 1. 📏

Solution 3: The definition of a polynomial states that variables must have “natural exponents,” which means non-negative integers (0, 1, 2, … and so on). In the expression x + x^(1/2):

• The first term x has an exponent of 1, which is a natural number. This is valid.
• The second term x^(1/2) has an exponent of 1/2. This is a rational exponent, not a natural (non-negative integer) exponent. Because of the x^(1/2) term, this expression does not qualify as a polynomial.