Question

State the conjugate of each binomial. $$3 t-8$$

Short answer

Question: Find the conjugate of the binomial \(3t-8\). Answer: The conjugate of the binomial \(3t-8\) is \(3t+8\).

Step-by-step solution

Identify the binomial

The given binomial is: $$3t - 8$$

Find the conjugate

To find the conjugate of the binomial, change the sign of its second term: $$3t \color{red}{+} 8$$ So, the conjugate of the binomial \(3t-8\) is: $$3t+8$$

Key concepts

Binomial

In prealgebra, a binomial is an algebraic expression consisting of exactly two terms. These terms are usually connected by either a plus (+) or a minus (−) sign. For the expression to count as a binomial, it must have two separate algebraic parts that are not combined, such as in the example \(3t - 8\). Here, "3t" and "8" are the two terms that make up the binomial.
Binomials can be part of larger expressions and play an important role in many algebraic operations.

• They are often used in polynomial expressions.
• They serve as the foundation for learning about quadratic expressions and equations.

Understanding how to work with binomials, like identifying their parts and recognizing their structure, is crucial as it provides a base for more complex mathematical concepts.

Conjugate

Finding the conjugate of a binomial is a simple yet significant operation in algebra. The conjugate changes the sign in the middle of the binomial's two terms. For instance, the conjugate of \(3t - 8\) is \(3t + 8\).
This operation is essential because it is often used to rationalize denominators and simplify expressions.

• By conjugating, we can eliminate radicals in the denominator of a fraction.
• It can also help in simplifying complex fractions by getting rid of imaginary numbers in the denominator.

The concept of a conjugate makes algebraic manipulation more streamlined and is a critical skill when working with expressions containing radicals or complex numbers.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Unlike an equation, it doesn’t contain an equals sign. They can vary in complexity from simple expressions like \(3 + 4\) to more intricate ones like \(3x^2 - 5x + 8\).
Algebraic expressions are fundamental in mathematics because they represent real-life situations and relationships.

• They can be used to model calculations and solve problems.
• They help in understanding and describing patterns.

Grasping how algebraic expressions work, their components, and operations like addition and subtraction between them is essential for advanced mathematics. One can manipulate and simplify these expressions using different rules and operations, paving the way to understanding equations and inequalities.