Question

State the conjugate of each binomial. $$-8 y-1$$

Short answer

Answer: The conjugate of the binomial \(-8y - 1\) is \(-8y + 1\).

Step-by-step solution

Identify the terms of the binomial

The given binomial is \(-8y - 1\). The two terms in this binomial are \(-8y\) and \(-1\).

Change the sign between the two terms

To find the conjugate of the binomial, we simply need to change the sign between the two terms. Since the given binomial is \(-8y - 1\), we will change the sign between the two terms to get the conjugate.

Write the conjugate of the binomial

After changing the sign between the two terms, we have the resulting binomial: \(-8y + 1\). So, the conjugate of the given binomial is \(-8y + 1\).

Key concepts

Algebra

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to solve equations and understand patterns.

In algebra, we often work with expressions that include variables, constants, and operations. These expressions can take many forms, ranging from simple equations like \(x + 2 = 5\) to complex formulas involving multiple variables and operations.

• Variables represent unknown numbers or quantities.
• Constants are fixed values, like \(2\) or \(-8\) in our example.
• Operations include addition, subtraction, multiplication, and division.

Understanding these components and how to manipulate them is key to mastering algebra. Changing the sign of a term, as seen in finding the conjugate, is a fundamental operation in this process.

Polynomials

Polynomials are expressions that consist of variables raised to different powers and multiplied by coefficients. They can have multiple terms, and each term is a product of a coefficient and a variable raised to an exponent.

For example, the polynomial \(-8y - 1\) is a binomial, which means it has two terms. In polynomials:

• Each term is a product of a number (the coefficient) and a variable raised to a power.
• The degree of a polynomial is determined by the highest exponent of the variable.
• When finding binomial conjugates, we change the sign between the terms.

Binomials are particularly useful for representing and solving quadratic equations and in various applications like physics and engineering.

Mathematical Expressions

Mathematical expressions are combinations of symbols that show a mathematical relationship or operation. They include numbers, operators, and variables, and they don’t always need to have an equation sign (\(=\)).

In expressions like \(-8y - 1\), we're looking at:

• Numbers, such as \(-8\) and \(-1\), which are coefficients and constants.
• Variables, like \(y\), which represent unknown quantities.
• Operators, including plus and minus, which indicate the mathematical operations to perform.

Expressions can be simplified, factored, or manipulated according to mathematical rules. Understanding these building blocks helps in tasks like finding conjugates, which involve flipping signs between terms to identify relationships and symmetries in mathematics.