Question

State the conjugate of each binomial. $$y+2$$

Short answer

Answer: The conjugate of the binomial $$y + 2$$ is $$y - 2$$.

Step-by-step solution

Identify the binomial

The given binomial is $$y + 2$$.

Change the sign of the second term

Change the sign of the second term from positive to negative. The conjugate of $$y + 2$$ is $$y - 2$$. The conjugate of the binomial $$y + 2$$ is $$y - 2$$.

Key concepts

Binomial Expressions

In algebra, a binomial expression is one of the simplest forms of an algebraic expression, consisting of two terms. These terms can be numbers, variables, or a combination of both, connected by a plus (\) or minus (\) sign. Examples of binomial expressions include \(x + y\) and \(a - 3\). When working with binomials, it's important to be familiar with certain operations, such as finding the conjugate.

The conjugate of a binomial is formed by changing the sign between the two terms, which is extremely helpful when multiplying binomials, factoring, and simplifying complex fractions. For instance, the conjugate of the binomial \(y + 2\) is \(y - 2\) as seen in the exercise. Mathematically, when a binomial is multiplied by its conjugate, the result is the difference of two squares, which is a vital concept in algebra.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can consist of numbers, variables, and arithmetic operations—addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions are the building blocks of algebra and are used to describe patterns, relationships, and changes. They can vary widely in complexity, from simple binomials, like \(y + 2\) or \(x - 5\', to more intricate multivariable and polynomial expressions.

Elements of Algebraic Expressions

Understanding the elements of algebraic expressions, including terms, coefficients, constants, and the proper use of symbols, is crucial in solving and manipulating these expressions to find solutions to equations or inequations. Variables in algebraic expressions represent unknown quantities and play a pivotal role as placeholders that can assume different values.

Arithmetic Operations in Algebra

The foundational arithmetic operations in algebra include addition, subtraction, multiplication, and division. These operations are extended from ordinary numbers to work with algebraic expressions. A clear understanding of how to properly execute these operations with variables is essential.

Operation Rules

Algebraic rules, such as the commutative, associative, and distributive properties, guide how to rearrange and combine like terms. For example, when multiplying a binomial by its conjugate, the distributive property allows us to expand the product to get a polynomial. Simplifying algebraic expressions often involves a combination of these operations, making complex problems more manageable by systematically breaking them down into simpler parts.