Problem 65 - Elementary and Intermediate Algebra

Question

List the like terms in each expression, if any. $$ -12 m^{4}-3 m^{3}+2 m^{2}-m^{3} $$

Step-by-Step Solution

Answer

The like terms are \(-3m^{3}\) and \(-m^{3}\).

Step 1: Understand the Expression

The expression given is \(-12m^{4} - 3m^{3} + 2m^{2} - m^{3}\). Our task is to list the like terms present in this expression.

Step 2: Identify the Like Terms

Like terms are terms that have the same variables raised to the same power. As we scan through the expression, we identify like terms. 1. \(-3m^{3}\) and \(-m^{3}\) are like terms because they both involve \(m^{3}\).2. \(-12m^{4}\) and \(2m^{2}\) do not have identical exponents with any other term, so they don’t have like terms in this expression.

Step 3: List the Like Terms

From our identification, the like terms in the expression are: \(-3m^{3}\) and \(-m^{3}\). So, we can say the like terms in the expression are: - \(-3m^{3}\) and \(-m^{3}\).

Key Concepts

Identifying Like Terms

In algebra, identifying like terms is an essential skill. Like terms are simply terms that have the exact same variable part, with both the variable itself and its exponent being identical. For example, in the expression

• -3m^{3}
• -m^{3}

these are like terms because they both contain the variable "m" raised to the power of 3. However, terms like

• -12m^{4}
• 2m^{2}

do not share the same exponents and therefore, are not considered like terms. Identifying like terms helps in simplifying algebraic expressions, as you can combine these terms by adding or subtracting their coefficients. This simplification is a building block for more advanced algebraic operations. When scanning expressions, always compare the variable and its exponent to determine likeness, not just the coefficients.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. An algebraic expression can be as simple as a single number or variable, or as complex as a multivariable polynomial. In the exercise, the expression

• -12m^{4} - 3m^{3} + 2m^{2} - m^{3}

is an excellent example of an algebraic expression. Each part of the expression separated by plus or minus signs is called a "term." The expression consists of four terms, some of which are like terms. Understanding algebraic expressions is key for simplifying equations and solving for unknowns. Remember, operations on expressions preserve equality when both sides of an equation are involved, which is crucial when solving algebraic equations.

Polynomials in Algebra

Polynomials are a special type of algebraic expression where the variables have whole number exponents. They consist of a sum of terms, each being a coefficient multiplied by a variable raised to an exponent. The expression given here is a polynomial:

• -12m^{4} - 3m^{3} + 2m^{2} - m^{3}

Polynomials are classified based on the number of terms they have:

• A monomial has one term.
• A binomial has two terms.
• A trinomial has three terms.
• Any polynomial with more than three terms is simply called a polynomial.

Our expression has four terms. The degree of a polynomial is determined by the highest exponent of its variables. In our case, that would be the term with exponent 4:

• -12m^{4}

Thus, it's a fourth-degree polynomial. Identifying the degree of polynomials is crucial for understanding their behavior and solving related algebra problems.