Problem 42 Write the opposite of the polyno... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 42

Write the opposite of the polynomial. $$12 x^{3}+4 x^{2}-8$$

Short Answer

Expert verified

The opposite polynomial is $-12x^3 - 4x^2 + 8$.

Step by step solution

- Understand the Concept

The opposite (or additive inverse) of a polynomial is obtained by changing the sign of every term in the polynomial.

- Identify the Terms

The polynomial given is: \[12x^3 + 4x^2 - 8\]Identify each term: $12x^3$, $4x^2$, and $-8$.

- Change the Signs

Change the sign of each term:- The term $12x^3$ becomes $-12x^3$.- The term $4x^2$ becomes $-4x^2$.- The term $-8$ becomes $8$.

- Write the Opposite Polynomial

Combine the terms with their new signs to write the opposite polynomial:\[-12x^3 - 4x^2 + 8\]

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Key Concepts

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

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can have constants, variables, and exponents, but only non-negative integer exponents.

For example, the polynomial given in the exercise is $12x^3 + 4x^2 - 8$. This includes coefficients (12, 4, -8) and variables (x) raised to integer powers (3 and 2).

Polynomials are fundamental in algebra and appear in various forms across mathematics. You can think of them as a way to represent continuous functions or to model different phenomena.

Additive Inverse

The concept of an additive inverse is crucial in mathematics. The additive inverse of a number is what you add to it to get zero. For example, the additive inverse of 5 is -5 because $5 + (-5) = 0$.

In the context of polynomials, an additive inverse is achieved by changing the sign of each term within the polynomial. This is clear from the original exercise:

Given the polynomial $12x^3 + 4x^2 - 8$, changing the signs of all the terms gives us its additive inverse: $-12x^3 - 4x^2 + 8$. Notice how each term has an opposite sign compared to the initial polynomial. This new polynomial will add to the original one to yield zero.

Changing Signs

Changing the signs of the terms in a polynomial is a straightforward but essential task. It involves flipping the sign before each term:

Positive terms become negative
Negative terms become positive

For instance, in the polynomial $12x^3 + 4x^2 - 8$:

The term $12x^3$ becomes $-12x^3$
The term $4x^2$ becomes $-4x^2$
The term $-8$ becomes $8$

This process allows us to find the additive inverse of the polynomial and is a fundamental skill for various algebraic manipulations. Essentially, it is about ensuring that every element in the polynomial is considered, and its sign is appropriately changed.

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91影视

Write the opposite of the polynomial. $$12 x^{3}+4 x^{2}-8$$

Short Answer

Step by step solution

- Understand the Concept

- Identify the Terms

- Change the Signs

- Write the Opposite Polynomial

Key Concepts

Polynomial

Additive Inverse

Changing Signs

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