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ZingPath: Polynomial Expressions and Factoring

Characteristics of Polynomials                     Searching for

Polynomial Expressions and Factoring

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Lesson Focus

Characteristics of Polynomials

Algebra-2

You will classify polynomials by degree and identify their terms, coefficients, standard form, and sums of coefficients.

Now You Know

After completing this tutorial, you will be able to complete the following:

• Explain what a polynomial is.
• Identify types of polynomial functions such as linear, quadratic, cubic, zero, and constant.
• Identify the terms in a polynomial function of one variable.
• Identify the coefficients of a polynomial function of one variable.
• Identify the degree of a polynomial function of one variable.
• Interpret and make use of the relation between the value of a polynomial and its constant term.
• Interpret and make use of the relation between the value of a polynomial and the sum of its coefficients.

Everything You'll Have Covered

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.

Tutorial Details

 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. Course Algebra-2 Type of Tutorial Concept Development Key Vocabulary coefficients, standard form, polynomials