Question

State the conjugate of each binomial. $$2 x+5$$

Short answer

Answer: The conjugate of the binomial \(2x+5\) is \(2x-5\).

Step-by-step solution

Identify the original binomial

The original binomial given is \(2x+5\).

Change the operation

Change the operation from addition to subtraction. We can do this by changing the plus sign to a minus sign while keeping the terms the same: \((2x + 5) \to (2x -5)\).

Write the conjugate

The conjugate of the given binomial \(2x+5\) is \(2x-5\).

Key concepts

Binomial

A binomial is a type of polynomial. It consists of exactly two terms which are usually connected by a plus or a minus sign. Here are a few important things to know about binomials:

• The two terms could be constants, variables, or a mix of both. For example, in the binomial \( 2x + 5 \), \(2x\) is a term with a variable, and \(5\) is a constant term.
• Operations between the terms can vary, commonly addition or subtraction.

Binomials are often used in algebra to form equations and can represent real-world scenarios. Understanding binomials is foundational, as they are building blocks to more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions consist of variables and constants, connected by operations such as addition, subtraction, multiplication, and sometimes division. They are the language of algebra and help us to model and solve mathematical problems. Here's what you need to know:

• Algebraic expressions can be as simple as a single number or variable. Examples include \(5\) or \(x\).
• They can also be more complex, like binomials or trinomials, which have two or three terms, respectively.
• When simplifying or transforming algebraic expressions, operations are performed according to the order of precedence, following rules like distributive, associative, and commutative laws.

Understanding how to manipulate and work with these expressions is key to solving algebraic equations. This includes finding conjugates in binomials, which involves swapping plus and minus signs between the terms.

Prealgebra

Prealgebra is an introductory course that lays the groundwork for all future algebraic learning. It bridges the gap between arithmetic and algebra. Some core concepts include:

• Basic arithmetic operations: addition, subtraction, multiplication, and division, all applied to both numbers and variables.
• Introduction to variables which are symbols used to represent numbers.
• Simple algebraic equations and expressions, including understanding binomials, which are key elements as you progress to more complex algebra.

Prealgebra serves as a stepping stone where basic mathematical ideas are expanded to include abstract thinking. This makes algebraic concepts, like understanding the conjugate of a binomial, easier as you advance in mathematical studies.