Problem 33 - College Algebra with Modeling and Visualization

Question

Find the opposite of the polynomial. $$z^{4}-z^{2}-9$$

Step-by-Step Solution

Answer

The opposite of the polynomial is \(-z^4 + z^2 + 9\).

Step 1: Understand the Concept of Opposite

The opposite of a polynomial is obtained by changing the sign of each term of the polynomial. For a term with a positive coefficient, we make it negative, and for a term with a negative coefficient, we make it positive.

Step 2: Apply the Concept to the Polynomial

The given polynomial is \( z^4 - z^2 - 9 \). We need to change the sign of each term:- The first term \( z^4 \) has a positive coefficient. Its opposite becomes \( -z^4 \).- The second term \( -z^2 \) has a negative coefficient. Its opposite becomes \( +z^2 \).- The third term \( -9 \) has a negative coefficient. Its opposite becomes \( +9 \).

Step 3: Write the Opposite Polynomial

After changing the signs, the opposite of the polynomial \( z^4 - z^2 - 9 \) is \( -z^4 + z^2 + 9 \).

Key Concepts

Polynomial Coefficients

In polynomials, coefficients are the numbers placed in front of the variables. Essentially, they act as multipliers for the terms of the polynomial. Understanding coefficients helps in many operations, including finding the opposite of a polynomial.

Let's consider the polynomial given in the exercise, which is expressed as \( z^4 - z^2 - 9 \). Here, the coefficients are:

• For \( z^4 \), the coefficient is \(+1\) because any term without an explicitly written number has a coefficient of \(+1\).
• For \(-z^2\), the coefficient is \(-1\).
• For \(-9\), considered a constant term, the coefficient is \(-9\).

Recognizing these coefficients allows us to switch their signs when finding the opposite of a polynomial. Coefficients play a pivotal role, so it's important to identify if they are positive or negative. This knowledge is vital for various algebraic processes.

Sign Change of Polynomials

Changing the sign of a polynomial involves flipping the signs of its coefficients. This transformation is central when determining what is known as the opposite of a polynomial. When working with the polynomial \( z^4 - z^2 - 9 \), changing the sign involves a straightforward process:

• The first term \( z^4 \), with a positive coefficient (+1), becomes \( -z^4 \).
• The second term \(-z^2\), with a negative coefficient (-1), transforms into \(+z^2\).
• The last term \(-9\), also negative, changes to \(+9\).

Making each sign change helps us find the opposite polynomial—another expression that is mathematically equivalent in magnitude but opposite in direction. Doing this provides us with \( -z^4 + z^2 + 9 \), clearly illustrating how simple sign changes can define an entirely new polynomial.

Algebraic Expressions

Polynomials are a subset of algebraic expressions, a broad category of mathematical statements consisting of numbers, variables, and arithmetic operations. These expressions can range from simple to complex.

The given polynomial \( z^4 - z^2 - 9 \) is an algebraic expression because it combines variables raised to powers and constants, all linked by subtraction. Polynomials like this are convenient for numerous mathematical tasks such as finding roots, factoring, and in this case, determining opposites.

To fully understand and manipulate algebraic expressions, mastering the basics like recognizing terms, coefficients, and operations is necessary. Knowing how each part of an algebraic expression works together is essential for advancing in mathematics. Concepts like simplifying, evaluating, and finding opposites truly highlight the dynamic nature of algebraic expressions.