What do you call the highest sum of the exponents of the variables in a polynomial?

The highest sum of the exponents of the variables in a polynomial is called the degree of the polynomial.

Understanding the Degree of a Polynomial

The "degree" of a polynomial helps us understand its behavior and characteristics. It's defined differently based on whether the polynomial has one variable or multiple variables.

Polynomials in One Variable

• Definition: The degree is the highest exponent of the variable.
• Example: In the polynomial 3x^4 + 2x^2 - x + 7, the degree is 4.

Polynomials in Multiple Variables

• Definition: First, find the sum of the exponents in each term. Then, the degree of the polynomial is the highest of these sums.

• Reference: As stated in the provided document, "In case of polynomials in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial."

• Example: Consider the polynomial 3x^2y^3 + 5xy^2 + x^4.

  Term	Sum of Exponents
  3x^2y^3	2 + 3 = 5
  5xy^2	1 + 2 = 3
  x^4	4

  The highest sum is 5, so the degree of the polynomial is 5.

Examples and Practical Insights

• Example 1: Find the degree of the polynomial 7a^3b^2c + 4a^2b + 9c^5

  Term	Sum of Exponents
  7a^3b^2c	3 + 2 + 1 = 6
  4a^2b	2 + 1 = 3
  9c^5	5

  Therefore, the degree of the polynomial is 6.

• Example 2: Determine the degree of 2x + 3y - 5.

  Term	Sum of Exponents
  2x	1
  3y	1
  -5	0

  Therefore, the degree is 1 (remember that a constant term has a degree of 0).