Understanding Monomials and Polynomials
This page introduces the fundamental concepts of monomials and polynomials, essential components of algebraic expressions. It provides definitions, examples, and key properties to help students grasp these important mathematical structures.
Definition: A monomial of degree n wherenisanon−negativeinteger in one real variable x is an expression that can be written in the form a·x^n, where a is a fixed real number not equal to zero.
Definition: A polynomial of degree n wherenisapositiveinteger in one real variable x is an expression that can be written in the form: a_n·x^n + a_n−1·x^n−1 + ... + a_1·x + a_0, where a_n, a_n−1, ..., a_1, a_0 are real numbers called the coefficients of the polynomial.
The page provides several examples to illustrate these concepts:
Example: Nx = 3x² - 2x⁵ + 4 is a polynomial with degree 5.
Example: Fx = -12x⁸ is a polynomial andalsoamonomial with degree 8.
Highlight: The degree of a polynomial is determined by the highest power of its variable term with a non-zero coefficient.
The text also mentions some important properties and special cases:
- The zero polynomial does not have a degree.
- The sum of all coefficients of a polynomial Wx is equal to W1.
- Constant polynomials polynomialsofdegree0 are discussed.
Vocabulary: Współczynniki wielomianu polynomialcoefficients are the numerical factors that multiply each power of the variable in a polynomial.
The page concludes with exercises that help reinforce the understanding of polynomial degrees and coefficients, including examples with parameters and special cases like √x + 8 = x + 8^1/2.
