Problem 32 - College Algebra with Modeling and Visualization

Question

Find the opposite of the polynomial. $$-x^{2}-x+6$$

Step-by-Step Solution

Answer

The opposite polynomial is \(x^2 + x - 6\).

Step 1: Identify the Polynomial

The given polynomial is \[-x^2 - x + 6\].Our objective is to find the opposite of this polynomial, which means we need to change the sign of each term.

Step 2: Change the Sign of Each Term

To find the opposite of the polynomial, we negate each term individually:1. The term \(-x^2\) becomes \(+x^2\).2. The term \(-x\) becomes \(+x\).3. The term \(+6\) becomes \(-6\).

Step 3: Combine the Opposite Terms

Combine the terms with their new signs to form the opposite polynomial:\[x^2 + x - 6\]

Key Concepts

Polynomial Negation

Polynomials are expressions consisting of variables and coefficients structured together using addition, subtraction, multiplication, and non-negative integer exponents. When we talk about polynomial negation, we mean finding the opposite polynomial, which involves changing the signs of all terms within the polynomial. For example, if we have a polynomial \(-x^2 - x + 6\), negating the polynomial means converting every term to its opposite sign:

• The term \(-x^2\) becomes \(x^2\).
• The term \(-x\) becomes \(x\).
• The term \(6\) becomes \(-6\).

Thus, when all the signs are flipped, \(-x^2 - x + 6\) turns into \(x^2 + x - 6\). This transformation is crucial for solving equations, simplifying expressions, or evaluating algebraic functions.

Terms in Algebra

In algebra, terms are essential building blocks. Each term in a polynomial can consist of:

• A coefficient, which is a numerical factor in front of the variable.
• A variable, which is an unknown quantity represented by letters like \(x\) or \(y\).
• An exponent, which indicates how many times the variable is multiplied by itself.

Understanding terms is fundamental for manipulating algebraic expressions. For instance, in the polynomial \(-x^2 - x + 6\), it contains three terms:

• First term: \(-x^2\)
• Second term: \(-x\)
• Third term: \(6\)

Each term must be handled individually when performing operations like addition, subtraction, and negation. Recognizing and separating these terms simplifies further algebraic operations, such as polynomial negation.

Sign Changes in Polynomials

Polynomials can have both positive and negative terms, defined by their signs. Changing signs in a polynomial is a common task, which is often necessary in mathematical contexts such as solving equations or performing polynomial negation. To change the sign of a polynomial term:

• Convert positive signs to negative signs and vice versa.
• Apply this change across each term in the polynomial.

In the given polynomial \(-x^2 - x + 6\), the sign of each term is switched to achieve the opposite effect, resulting in \(x^2 + x - 6\):

• From \(-x^2\) to \(x^2\)
• From \(-x\) to \(x\)
• From \(6\) to \(-6\)

This transformation reorients the polynomial in an algebraically meaningful way, which could change its roots or its graphical representation. Understanding sign changes helps in recognizing how polynomials behave under different operations and conditions.