Problem 29 - College Algebra with Modeling and Visualization

Question

Find the opposite of the polynomial. $$7 x^{3}$$

Step-by-Step Solution

Answer

The opposite polynomial is \(-7x^{3}\).

Step 1: Understanding the Problem

The given task is to find the opposite of the polynomial. The given polynomial is \(7x^{3}\). To find the opposite, we need to change the sign of every term in the polynomial.

Step 2: Identify the Term

The polynomial given is \(7x^{3}\), which is a single term, also known as a monomial. Since it consists of only one term, we will focus on changing the sign of this particular term.

Step 3: Change the Sign

To find the opposite of the polynomial, change the sign of the coefficient. The coefficient of \(x^{3}\) is 7. Changing the sign of 7 gives us -7. Therefore, the opposite term is \(-7x^{3}\).

Step 4: Write the Opposite Polynomial

After changing the sign of the term, the opposite of the original polynomial \(7x^{3}\) becomes \(-7x^{3}\). This is the final answer.

Key Concepts

Monomials

A monomial is a mathematical expression that consists of only one term. These are simple expressions that can include numbers, variables, or a combination of both. For example, in our given exercise, the term \(7x^3\) is a monomial. It contains only two parts: a coefficient (7) and a variable term (\(x^3\)).
Monomials are essential building blocks of more complex polynomial expressions. They can be thought of as the simplest type of polynomial. Unlike a polynomial with multiple terms, a monomial does not have any addition or subtraction within itself.
Understanding monomials is the first step to mastering more complex polynomial operations. Recognizing them quickly helps in solving problems like finding the opposite of a polynomial, where focusing on the sign of the monomial is key.

Coefficients

Coefficients are the numerical part in front of the variables in algebraic expressions such as monomials. They are crucial since they dictate the scale or magnitude of a term. In the polynomial \(7x^3\), 7 is the coefficient. It signifies that \(x^3\) is scaled by 7 times its inherent value.
When dealing with algebraic expressions, knowing how to manipulate coefficients is fundamental. For instance, finding the opposite of a polynomial involves changing the sign of the coefficient in each term.

• The coefficient affects the steepness and position of the graph of a polynomial function.
• It plays a key role in determining the arithmetic operations on polynomials, like addition, subtraction, and multiplication.

Changing the sign of a coefficient transforms the term accordingly, and in our example, the positive 7 becomes a negative 7, forming \(-7x^3\).

Polynomial Sign Change

Changing the sign of a polynomial is a straightforward yet significant operation in algebra. The process involves inverting the sign of each term's coefficient in a given polynomial.
For instance, if you have a positive coefficient, you make it negative, and vice versa. This operation results in what is often referred to as the 'opposite' of the polynomial.
In the case of a monomial like \(7x^3\), the sign change involves reversing the sign of its single coefficient. So, the given polynomial's opposite is \(-7x^3\).
It's important to adapt quickly between polynomials and their opposites, as this skill is frequently utilized in various algebraic operations like subtraction of polynomials, solving equations, and finding zeros of functions.
Effectively applying polynomial sign changes can simplify problems and lead to swifter solutions.