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Chapter 10: Problem 44

Write the opposite of the polynomial. $$2.4 m-5.1 n$$

Short Answer

Expert verified
\(-2.4m + 5.1n\)

Step by step solution

01

Identify the Given Polynomial

The polynomial provided is \(2.4m - 5.1n\).
02

Understand What 'Opposite' Means

'Opposite' of a polynomial means to change the sign of each term in the polynomial. For instance, \(a\) becomes \(-a\), and \(-b\) becomes \(b\).
03

Change the Sign of Each Term

The term \(2.4m\) changes to \(-2.4m\), and the term \(-5.1n\) changes to \(5.1n\).
04

Write the Opposite Polynomial

Combining the terms, the opposite polynomial is \(-2.4m + 5.1n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial
A polynomial is an algebraic expression that consists of variables and coefficients. Polynomials are composed of terms that are added or subtracted. These terms are usually made up of the product of a number (called the coefficient) and a variable that has been raised to a power. For example, in the expression \(3x^2 - 4x + 7\), the terms are \(3x^2\), \(-4x\), and \(7\). Polynomials can vary in complexity from simple (like \(x + 2\)) to more complex (like \(4a^3 - 3a^2 + 2a - 1\)). Recognizing the components of a polynomial helps in understanding how to perform operations like addition, subtraction, and finding the opposite.
changing signs
When you are asked to find the opposite of a polynomial, you need to change the sign of each term. In other words, if a term is positive, it becomes negative, and if it is negative, it becomes positive. Let's take a closer look:
  • If you have the term \(2.4m\), its opposite would be \(-2.4m\).
  • If you have \(-5.1n\), changing the sign would make it \(5.1n\).
This change of signs is straightforward but essential for finding the opposite polynomial. By reversing the signs of each term, you transform the equation completely, reflecting it across the origin in a graph.
terms in equations
Understanding terms in an equation is crucial for manipulating polynomials correctly. Each 'term' in a polynomial is a part of the expression that is separated by a plus or minus sign. For example, in the original exercise polynomial \(2.4m - 5.1n\), there are two terms:
  • \(2.4m\) - The first term, composed of the coefficient \(2.4\) and the variable \(m\).
  • \(-5.1n\) - The second term, which has the coefficient \(-5.1\) and the variable \(n\).
When finding the opposite of a polynomial, it's important to consider each term individually to change its sign. Each term's sign change plays a critical role in rewriting the expression correctly. Combining these modified terms gives you the new polynomial, representing its opposite.

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