Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Polynomials Intro

College Algebra

0. Review of Algebra

Polynomials Intro

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1:39 minutes

Problem 19

Textbook Question

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. (3/8)x^5-(1/x^2)+9

Verified step by step guidance

Step 1: Identify the terms in the expression: \( \frac{3}{8}x^5 \), \( -\frac{1}{x^2} \), and \( 9 \).

Step 2: Check if each term is a polynomial term. A polynomial term has variables raised to non-negative integer powers.

Step 3: The term \( \frac{3}{8}x^5 \) is a polynomial term because the variable \( x \) is raised to the power of 5, which is a non-negative integer.

Step 4: The term \( -\frac{1}{x^2} \) is not a polynomial term because the variable \( x \) is in the denominator, equivalent to \( x^{-2} \), which is a negative power.

Step 5: Since not all terms are polynomial terms, the entire expression is not a polynomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Definition

A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Each term in a polynomial is formed by multiplying a coefficient by a variable raised to a power. Expressions that include negative exponents, fractional exponents, or variables in the denominator are not considered polynomials.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It provides insight into the polynomial's behavior and shape when graphed. For example, in the polynomial 3x^4 + 2x^3 - x + 5, the degree is 4, as the term with the highest exponent is x^4.

Classification of Polynomials

Polynomials can be classified based on the number of terms they contain. A monomial has one term, a binomial has two terms, and a trinomial has three terms. If a polynomial has more than three terms, it is simply referred to as a polynomial. This classification helps in understanding the structure and complexity of the polynomial.