Problem 31 - College Algebra with Modeling and Visualization

Question

Find the opposite of the polynomial. $$19 z^{5}-5 z^{2}+3 z$$

Step-by-Step Solution

Answer

The opposite of the polynomial is \(-19z^5 + 5z^2 - 3z\).

Step 1: Understanding the Problem

To find the opposite of a polynomial, you need to find the additive inverse of each term in the polynomial.

Step 2: Identify the Terms of the Polynomial

The given polynomial is: \(19z^{5} - 5z^{2} + 3z\). The terms here are \(19z^{5}\), \(-5z^{2}\), and \(+3z\).

Step 3: Calculate the Opposite of Each Term

The opposite of \(19z^5\) is \(-19z^5\). The opposite of \(-5z^2\) is \(+5z^2\). The opposite of \(+3z\) is \(-3z\).

Step 4: Write the Opposite Polynomial

Combining the opposites of each term, we get the opposite polynomial: \(-19z^5 + 5z^2 - 3z\).

Key Concepts

Additive Inverse

The concept of an additive inverse is fundamental in mathematics. It refers to a value that, when added to the original number, yields zero. In terms of polynomials, finding the additive inverse involves changing the sign of each term in the polynomial.

• If you have a positive term, like \(+5\), its additive inverse would be \(-5\).
• Conversely, if the term is negative, such as \(-7\), the additive inverse is \(+7\).

Understanding the additive inverse helps in simplifying expressions and equations, especially when you're attempting to eliminate terms or solve for zero. So, whenever you need to "find the opposite," think about switching the signs of each polynomial term.

Polynomial Terms

A polynomial consists of terms, each of which is a product of a constant and a variable raised to a non-negative integer power. In our example, the given polynomial is composed of three distinct terms:

• \(19z^5\), where \(19\) is the coefficient and \(z^5\) indicates the variable part.
• \(-5z^2\), with \(-5\) as the coefficient and \(z^2\) as the variable element.
• \(+3z\), having \(+3\) as the coefficient and \(z\) as the variable part.

Each term in a polynomial is distinct due to its different degree, normally determined by the power of the variable (like the 5 in \(z^5\)). Understanding and identifying these terms is crucial for manipulating polynomials, whether you're adding, subtracting, or finding the opposite.

Opposite Polynomial

An opposite polynomial is essentially the result of applying the concept of additive inverse across all the terms in a polynomial. Once you've understood each term and its additive inverse, creating the opposite polynomial becomes straightforward.
For the polynomial \(19z^5 - 5z^2 + 3z\), the opposite polynomial involves the following changes:

• \(19z^5\) becomes \(-19z^5\).
• \(-5z^2\) becomes \(+5z^2\).
• \(+3z\) turns into \(-3z\).

Combine these opposite terms, and the final opposite polynomial is \(-19z^5 + 5z^2 - 3z\). This process reverses the direction of the numbers' values in terms of their sign, effectively reflecting the polynomial around the zero point, much like finding the negative of a number.