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Chapter 10: Problem 41

Write the opposite of the polynomial. $$9 x^{2}-2 x-3$$

Short Answer

Expert verified
-9x^2 + 2x + 3

Step by step solution

01

Understanding the problem

To write the opposite of the polynomial, each term of the polynomial must have its sign reversed.
02

Identify each term

The polynomial given is: \[9x^2 - 2x - 3\].Identify the three terms: \[9x^2, -2x, \text{and} -3.\]
03

Reverse the signs

Change the sign of each term in the polynomial: \[9x^2\rightarrow -9x^2,\] \[-2x \rightarrow +2x,\] \[-3 \rightarrow +3.\]
04

Write the new polynomial

Combine the new terms to write the opposite polynomial: \[-9x^2 + 2x + 3.\]

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Key Concepts

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

Polynomial Terms
Polynomials are expressions composed of terms. Each term contains a variable raised to a power and may include a coefficient. For instance, in \[9x^2 - 2x - 3\], there are three terms: \[9x^2\], \[-2x\], and \[-3\]. The variable in our example is \['x'\], and each term combines the variable and its coefficient (the number next to the variable). Understanding terms is crucial because it helps in identifying the components that need any particular mathematical operation, such as reversing signs or combining like terms.
Sign Reversal
To find the opposite of a polynomial, we need to reverse the sign of each of its terms. Let's break down how to reverse each term:
  • The term \[9x^2\] becomes \[-9x^2\].
  • The term \[-2x\] becomes \[+2x\].
  • The term \[-3\] becomes \[+3\].
Sign reversal simply changes a positive sign to a negative sign, and a negative sign to a positive sign. This step-by-step method helps in accurately determining the opposite polynomial.
Combining Like Terms
Combining like terms is an important step in simplifying polynomials, though it isn't used in the specific exercise of finding the opposite polynomial. Nevertheless, understanding this concept enhances your algebra skills. Like terms are terms that have the same variable raised to the same power. For example, \[5x^2\] and \[-3x^2\] are like terms because they both have \['x'\] raised to the power of 2. When combining like terms, add or subtract the coefficients and keep the variable part unchanged. For instance: \[(5x^2 - 3x^2) = 2x^2\]. This process simplifies expressions and helps in solving polynomial equations efficiently.

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Most popular questions from this chapter

Evaluate the polynomial for the given values of the variable. \(t^{2}-6 t-1\) a. for \(t=-2\) b. for \(t=\frac{1}{4}\)

Simplify the expression. $$\left(-4 a^{5}\right)^{2}(a b)^{3}$$

Subtract the polynomials. $$\begin{array}{r}15 a^{3}-2 a^{2}+4 a \\\\-\left(2 a^{3}+6 a^{2}-12 a\right) \\\\\hline\end{array}$$

Simplify. Write the answers with positive exponents only. $$\frac{m^{5}}{m^{7}}$$

Simplify. Write the answer with positive exponents only. $$\frac{2}{x^{-2} y^{-3}}$$
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