Full Prep Academy

Overview

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. Polynomials are used in many areas of mathematics and science to describe a wide range of phenomena.

Key Terminology

To better understand polynomials, it's essential to familiarize yourself with the following terms:

• Term: Each individual part of a polynomial, separated by a plus (+) or minus (-) sign.
• Coefficient: The numerical factor of each term.
• Degree: The highest power of the variable in a polynomial.

Types of Polynomials

Polynomials can be classified based on the number of terms:

• Monomial: A polynomial with one term, e.g., \(3x\).
• Binomial: A polynomial with two terms, e.g., \(x + 5\).
• Trinomial: A polynomial with three terms, e.g., \(x^2 + 3x - 4\).

Example: Identifying Terms, Coefficients, and Degree

Consider the polynomial expression:

In this example:

• Terms: 4x³, -2x, 7
• Coefficients: 4, -2
• Degree: 3 (highest exponent)

Practice Questions

1. Question 1: Identify the terms, coefficients, and degree of the polynomial: \( 5x^4 - 3x^2 + x - 6 \)
   Solution

   Terms: 5x⁴, -3x², x, -6

   Coefficients: 5, -3, 1

   Degree: 4

2. Question 2: What type of polynomial is the expression: \( x^2 + 7x \)
   Solution

   This is a binomial because it has two terms.

3. Question 3: Identify the degree of the polynomial: \( 2a^3 + 4a - 1 \)
   Solution

   The degree is 3, as 3 is the highest exponent in the polynomial.

4. Question 4: Which of the following is a trinomial?
   (a) \(3x + 2\)
   (b) \(x^2 - 5x + 4\)
   (c) \(6y\)
   (d) \(4x^3 - x\)
   Solution

   (b) is a trinomial because it contains three terms: \(x^2\), \(-5x\), and \(4\).

5. Question 5: Identify the coefficients in the polynomial: \( -7x^3 + 5x^2 - x + 8 \)
   Solution

   The coefficients are -7, 5, -1, and 8 (for the constant term).