You will classify polynomials by degree and identify their terms, coefficients, standard form, and sums of coefficients.
After completing this tutorial, you will be able to complete the following:
A polynomial P(x) of degree n with real coefficients is a mathematical expression of the form are real numbers, , and n is a nonnegative integer. The term is called the leading term, whereas is called the constant term. Polynomials are classified in two ways: by the number of terms and by the degree of the polynomial.
Classification by number of terms:
· monomial - a polynomial with one term (e.g., :4x , -1, 3x^4 )
· binomial - a polynomial with two terms (e.g., : 4x-3, 3x^4 +x)
· trinomial - a polynomial with three terms (e.g., : -3x^3-x+6, 3x^3 +x^2+6 )
· polynomial - the general name for a polynomial with 1 or more terms (e.g., 3x^4+x+2x^3+x^2-4x-9, 3x^3+x^2+6)
Classification of polynomials by degree:
· linear - a polynomial of degree one (e.g., : 2x)
· quadratic - a polynomial of degree two (e.g., : x^2+4 )
· cubic - a polynomial of degree three (e.g., : 3x^3+x^2+6)
· quartic - a polynomial of degree four (e.g., : x4 - 5)
· quintic - a polynomial of degree five (e.g., : x5 - 2x4 - 3x + 5)
Polynomials with a degree higher than three are also called by their numeric degree; for example, x^5 ? 2x^4 + 5x^3 +x^2 -x +6 can be referred to as a fifth degree polynomial.
Characteristics of polynomials.
Each polynomial has a leading term and a leading coefficient. Polynomials also have a constant term.
· leading term - the term with the greatest exponent of x; for example, in the polynomial f(x) = 5x4 - x3 + 6x - 10, the leading term is 5x4.
· leading coefficient - the number multiplying the variable in the leading term; in the polynomial f(x) = 5x4 - x3 + 6x - 10, the leading term is 5x4; the leading coefficient is 5.
· constant term - a term with degree 0; in the example f(x) = 5x4 - x3 + 6x - 10, the term with degree zero is -10 since 10 = 10x0.
The constant term of a polynomial gives the y-intercept of the graph. Recall that the x-coordinate of any point along the y-axis is 0, so the y-intercept of a function occurs when x = 0. So, for a polynomial function, when evaluated at x = 0, we get
For example, the constant term of f(x) = x2 - 1 - 1, and the graph crosses the y-axis at (0, -1).
Standard form of a polynomial.
The standard form of a polynomial is . Any polynomial can be written in standard form by combining like terms and writing the terms in order of descending degree. For example, the standard form of 1-x+x^2-4x is given by -x^2-5x+1. The leading term, constant term, and degree of a polynomial are easy to read from the standard form.
|Approximate Time||30 Minutes|
|Pre-requisite Concepts||Learners should understand the concept of a function and function notation; identify the terms, coefficients, and leading coefficient of an algebraic expression; and simplify algebraic expressions using the properties of exponents.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||coefficients, standard form, polynomials|