Polynomial Definition Practice

Definition of a Polynomial:

1. A polynomial in x (or some other variable such as y, z,...) is defined as:
   • A single term or a finite sum of terms of the form axⁿ where:
     • a is a real number
     • n is a whole number {0, 1, 2, 3,...}
2. An algebraic expression is NOT a polynomial when any of the following apply:
   • A term contains a variable in the denominator or a variable with a negative exponent:
     • $$\frac{1}{x} = x^{-1}$$
     • $$x^{-5} = \frac{1}{x^{5}}$$
   • A term contains a variable with a fractional exponent or a variable under a radical:
     • $$\large{\sqrt{x} = x^{\frac{1}{2}}}$$
     • $$\large{x^{\frac{2}{3}} = \sqrt[3]{x^{2}}}$$

Classifying a Polynomial:

1. A polynomial with only one term is known as a monomial
   • Ex: 5x²
   • Ex: 3
2. A polynomial with only two terms is known as a binomial
   • Ex: -4x³ - 5
   • Ex: 3x² + 1
3. A polynomial with only three terms is known as a trinomial
   • Ex: 9x² + 4x - 2
   • Ex: 14x³ - 11x² + 8
4. A polynomial with more than three terms does not have a special name
   • Ex: 5x³ - 2x² + x + 5
   • Ex: -9x⁴ - 8x³ + 12x² + 7x - 1

Step-by-Step: