what is a degree of a polynomial

Daniel Parker

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15-03-2025

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Polynomials are fundamental building blocks in algebra. Understanding their properties, especially the degree, is crucial for further mathematical studies. This article will clearly explain what the degree of a polynomial is and how to find it. We'll cover various examples to solidify your understanding.

Understanding Polynomials

Before diving into the degree, let's briefly refresh what a polynomial is. A polynomial is an expression consisting of variables (often represented by 'x'), coefficients, and exponents, combined using addition, subtraction, and multiplication. Crucially, there are no division by variables, and exponents are always non-negative integers.

Here are some examples of polynomials:

• 3x² + 2x - 5
• 5x⁴ - 2x² + 7
• x + 1
• 8 (a constant is also considered a polynomial)

These are not polynomials:

• 1/x (division by a variable)
• x⁻² (negative exponent)
• √x (fractional exponent)

Defining the Degree of a Polynomial

The degree of a polynomial is the highest power (exponent) of the variable in the polynomial. It indicates the polynomial's complexity.

Let's look at examples to illustrate:

Example 1:

3x² + 2x - 5

The highest power of 'x' is 2. Therefore, the degree of this polynomial is 2. This is also called a quadratic polynomial.

Example 2:

5x⁴ - 2x² + 7

The highest power of 'x' is 4. The degree of this polynomial is 4. This is a quartic polynomial.

Example 3:

x + 1

The highest power of 'x' is 1 (remember, x is the same as x¹). Therefore, the degree is 1. This is a linear polynomial.

Example 4:

8

A constant term like 8 can be considered as 8x⁰. Since the power of x is 0, the degree of a constant polynomial is 0.

Example 5: Polynomials with Multiple Variables

The concept of degree extends to polynomials with multiple variables. In such cases, the degree is the highest sum of exponents in any single term.

For example, consider 2x³y² + 5xy⁴ - 7x²y.

• In the term 2x³y², the sum of exponents is 3 + 2 = 5.
• In the term 5xy⁴, the sum of exponents is 1 + 4 = 5.
• In the term -7x²y, the sum of exponents is 2 + 1 = 3.

Therefore, the degree of this polynomial is 5 (the highest sum of exponents).

Why is the Degree Important?

The degree of a polynomial has several significant implications:

• Graphing: The degree influences the shape and number of turning points in the graph of the polynomial function.
• Number of Roots: The Fundamental Theorem of Algebra states that a polynomial of degree 'n' has exactly 'n' roots (including complex roots).
• Polynomial Operations: Understanding the degree helps predict the degree of the resulting polynomial when adding, subtracting, or multiplying polynomials. For example, adding two polynomials of degree 3 will result in a polynomial of degree at most 3.

Conclusion: Mastering Polynomial Degrees

Understanding the degree of a polynomial is a cornerstone of algebra. By mastering this concept, you'll be better equipped to handle more advanced topics in mathematics and related fields. Remember to always identify the highest power of the variable (or the sum of exponents in multivariable cases) to determine the degree of any given polynomial.