Classify The Following Polynomials. Combine Any Like Terms First.

Classifying Polynomials: A Comprehensive Guide

Understanding polynomials is fundamental to algebra and many other branches of mathematics. This article will delve into the classification of polynomials, focusing on combining like terms before classification. We’ll explore different types of polynomials, their characteristics, and provide numerous examples to solidify your understanding. By the end, you'll be able to confidently classify any polynomial you encounter.

What is a Polynomial?

A polynomial is an expression consisting of variables (often denoted by x, y, z, etc.) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A single term within a polynomial is called a monomial. Multiple monomials combined form a polynomial.

Examples of Polynomials:

• 3x² + 2x - 5
• 7y⁴ - 2y² + 1
• 5x³y² + 2xy - 4
• 10 (This is a constant polynomial)
• x (This is a linear monomial)

Examples of Expressions that are NOT Polynomials:

• 2/x (Negative exponent)
• √x (Fractional exponent)
• x⁻² + 4 (Negative exponent)
• 5xˣ (Variable in the exponent)

Combining Like Terms: A Crucial First Step

Before classifying a polynomial, it's essential to simplify it by combining like terms. Like terms are terms that have the same variables raised to the same powers. This simplification makes the classification process much easier and more accurate.

Example:

Let's consider the polynomial: 3x³ + 2x² - 5x + 7x² + x³ - 2x + 8

1. Identify Like Terms:

   • 3x³ and x³ are like terms.
   • 2x² and 7x² are like terms.
   • -5x and -2x are like terms.
   • 8 is a constant term (it's a like term to itself).

2. Combine Like Terms:

   3x³ + x³ = 4x³ 2x² + 7x² = 9x² -5x - 2x = -7x The constant term remains 8.

3. Simplified Polynomial:

The simplified polynomial is 4x³ + 9x² - 7x + 8. This is much easier to classify than the original, unsimplified form.

Classifying Polynomials: Based on Degree and Number of Terms

Polynomials are classified in two primary ways:

• By Degree: The degree of a polynomial is determined by the highest power of the variable in the polynomial.

• By Number of Terms: The number of terms (monomials) in the polynomial dictates its name.

Let's examine each classification in detail:

Classification by Degree:

• Constant Polynomial: A polynomial with degree 0. It has only a constant term. Example: 5, -2, 100.

• Linear Polynomial: A polynomial with degree 1. It contains a variable raised to the power of 1. Example: 2x + 3, y - 7, x.

• Quadratic Polynomial: A polynomial with degree 2. The highest power of the variable is 2. Example: 3x² + 2x - 1, y² + 5, -x² + 7x.

• Cubic Polynomial: A polynomial with degree 3. The highest power of the variable is 3. Example: x³ - 4x² + x + 6, 2y³ + y - 1.

• Quartic Polynomial: A polynomial with degree 4. Example: 5x⁴ - 3x² + x + 2.

• Quintic Polynomial: A polynomial with degree 5. Example: x⁵ + 2x³ - x² + 1.

Polynomials with a degree higher than 5 are generally referred to as polynomials of degree n (where n is the degree).

Classification by Number of Terms:

• Monomial: A polynomial with only one term. Example: 3x², -5y³, 7.

• Binomial: A polynomial with two terms. Example: 2x + 3, x² - 4, 3y³ + 7y.

• Trinomial: A polynomial with three terms. Example: x² + 2x - 5, 2y³ - y + 1.

Polynomials with four or more terms are simply called polynomials. There are no specific names for polynomials with more than three terms.

Putting it All Together: Examples

Let's classify some polynomials after combining like terms:

Example 1:

5x³ + 2x - 7x³ + 4x + 6

1. Combine Like Terms: 5x³ - 7x³ = -2x³, 2x + 4x = 6x. The simplified polynomial is -2x³ + 6x + 6.

2. Classification:

   • Degree: 3 (cubic)
   • Number of Terms: 3 (trinomial)

Therefore, this is a cubic trinomial.

Example 2:

2x²y + 3xy² - 5x²y + xy² + 1

1. Combine Like Terms: 2x²y - 5x²y = -3x²y, 3xy² + xy² = 4xy². The simplified polynomial is -3x²y + 4xy² + 1.

2. Classification:

   • Degree: 3 (The highest sum of exponents is 2+1=3)
   • Number of Terms: 3 (trinomial)

This is a 3rd degree trinomial. Note that the degree is determined by the highest sum of the exponents in any individual term.

Example 3:

4x⁴ - 2x² + 7x⁴ - 5x + 3

1. Combine Like Terms: 4x⁴ + 7x⁴ = 11x⁴. The simplified polynomial is 11x⁴ - 2x² - 5x + 3.

2. Classification:

   • Degree: 4 (quartic)
   • Number of Terms: 4 (polynomial)

This is a quartic polynomial.

Example 4:

6 - 2x + x² + 3x - 9

1. Combine Like Terms: 6 - 9 = -3, -2x + 3x = x. The simplified polynomial is x² + x - 3.

2. Classification:

   • Degree: 2 (quadratic)
   • Number of Terms: 3 (trinomial)

This is a quadratic trinomial.

Example 5: 7xy + 2xz - 3xy + 5xz

1. Combine Like Terms: 7xy - 3xy = 4xy, 2xz + 5xz = 7xz. The simplified polynomial is 4xy + 7xz.

2. Classification:

   • Degree: 2 (The highest sum of exponents in a term is 1 + 1 = 2)
   • Number of Terms: 2 (binomial)

This is a second-degree binomial.

Multivariable Polynomials

The examples above mostly involved single variables. However, polynomials can contain multiple variables. The degree of a multivariable polynomial is found by identifying the term with the highest sum of exponents.

Conclusion

Classifying polynomials is a fundamental skill in algebra. Remember that combining like terms is crucial before classifying. By understanding the concepts of degree and number of terms, you can confidently classify any polynomial, whether it's a simple monomial or a complex multivariable expression. Mastering this skill lays a solid foundation for tackling more advanced algebraic concepts. Practice consistently with various examples to improve your proficiency.