What is the Degree of a Polynomial Function Equation?

The degree of a polynomial function equation is the highest power of the variable in the polynomial.

To understand this, let's break it down:

• Polynomial Function: An expression containing variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

• Term: A single part of the polynomial separated by addition or subtraction. For example, in the polynomial 3x^2 + 2x - 5, 3x^2, 2x, and -5 are terms.

• Degree of a Term: The sum of the exponents of the variables in that term. For example:

  • The degree of 3x^2 is 2.
  • The degree of 2x (which is 2x^1) is 1.
  • The degree of -5 (which is -5x^0) is 0.
  • The degree of 4x^3y^2 is 3 + 2 = 5.

• Degree of the Polynomial: The highest degree of any term in the polynomial.

Examples:

• f(x) = 5x^3 - 2x + 1: The highest power of x is 3, so the degree is 3.

• g(x) = x - 7: The highest power of x is 1 (since x = x^1), so the degree is 1. This is also known as a linear function.

• h(x) = 9: This can be written as h(x) = 9x^0. The degree is 0. This is a constant function.

• k(x) = 2x^4 - x^2 + 3x^5 - 7: The highest power of x is 5, so the degree is 5. (Note that the terms don't have to be written in descending order of exponents).

• l(x, y) = 3x^2y + 5xy^3 - x + 2y - 8: The term 5xy^3 has degree 1 + 3 = 4, which is the highest among all terms, so the degree is 4.

Key Points:

• The degree of a polynomial is always a non-negative integer.
• The degree of a polynomial tells you something about the function's behavior, particularly its end behavior (what happens to the function as x approaches positive or negative infinity).

In summary, the degree of a polynomial function is determined by the highest exponent of its variable(s) after the polynomial has been simplified.