Problem 66 - Elementary and Intermediate Algebra

Question

List the like terms in each expression, if any. $$ 6 x^{3}+3 x^{2}+6 x $$

Step-by-Step Solution

Answer

There are no like terms in the expression.

Step 1: Identify Terms

First, we need to identify all the terms in the expression: \(6x^{3}\), \(3x^{2}\), and \(6x\).

Step 2: Determine the Degree of Each Term

Determine the degree of each term by looking at the exponent of the variable: The degree of \(6x^{3}\) is 3, \(3x^{2}\) is 2, and \(6x\) is 1.

Step 3: Group Like Terms

Like terms have the same variable raised to the same power. Here, each term has a different degree (3, 2, 1), so there are no like terms.

Key Concepts

Terms in Algebra

When dealing with algebraic expressions, terms are the building blocks that shape our understanding of the expression itself. In algebra, a term is a distinct component separated by a plus or minus sign. Each term is composed of a coefficient, a variable, and a power. For instance, in the expression \(6x^3 + 3x^2 + 6x\), we have three terms.

• The first term is \(6x^3\)
• The second term is \(3x^2\)
• The third term is \(6x\)

These terms consist of numbers and variables. The number in front of a variable, like 6 in \(6x^3\), is known as the coefficient. The letter "x" represents the variable, and the exponent (the small number above the variable) represents the power to which the variable is raised. Understanding terms is vital when manipulating and simplifying algebraic expressions. Think of terms as the pieces of a jigsaw puzzle where each piece plays a crucial role in the big picture.

Polynomial Degree

Understanding the degree of a term or an expression is crucial in algebra, especially when working with polynomials. The degree of a term is the sum of the exponents of its variables. If a term has just one variable, like \(6x^3\), the degree is simply the exponent of that variable.The expression \(6x^3 + 3x^2 + 6x\) contains terms of varying degrees:

• \(6x^3\) has a degree of 3.
• \(3x^2\) has a degree of 2.
• \(6x\) has a degree of 1.

For a whole polynomial, the highest degree among its terms determines the degree of the entire polynomial. Here, the highest degree is 3, so the polynomial is a cubic polynomial. Knowledge of polynomial degrees helps in understanding the behavior of expressions and can play a role in solving equations and graphing functions.

Like Terms

When studying algebraic expressions, an important concept is that of like terms. Like terms are terms within an expression that have the same variable raised to the same power.In terms of coefficients:- The coefficients do not need to be the same for terms to be considered like terms.- Only the variable and the exponent must match.For instance, in the expression \(6x^3 + 3x^2 + 6x\), each term has a different exponent:

• The first term, \(6x^3\), has an exponent of 3.
• The second term, \(3x^2\), has an exponent of 2.
• The third term, \(6x\), has an exponent of 1.

Since none of these terms share the same variable and power combination, they are not like terms. Recognizing like terms is crucial when simplifying expressions, as only like terms can be added or subtracted from one another. This simplifies the expression and can make it easier to work with in further calculations.