- Distribute the negative sign to all terms within the second polynomial.
- Group the terms by the power of the variable \(m\).
- Add or subtract the coefficients for each corresponding degree of \(m\).
- State the final simplified result: \(\boxed{-5 m^3 + m^2 + \frac{2}{9} m}\)
Explanation
- Setting up the subtraction
To find the difference between the two polynomials, we begin by setting up the subtraction:
\[ \left(\frac{8}{9} m^2 + \frac{1}{3} m\right) - \left(5 m^3 - \frac{1}{9} m^2 + \frac{1}{9} m\right) \]
Our objective is to simplify this expression by first removing the parentheses. This requires distributing the negative sign to each term inside the second set of parentheses. This transformation gives:
\[ \frac{8}{9} m^2 + \frac{1}{3} m - 5 m^3 + \frac{1}{9} m^2 - \frac{1}{9} m \]
- Grouping like terms
Next, we group the like terms together. Like terms are terms that have the same variable raised to the same power. Here, we identify the terms with \(m^3\), \(m^2\), and \(m^1\):
- For \(m^3\): \(-5 m^3\)
- For \(m^2\): \(\left(\frac{8}{9} m^2 + \frac{1}{9} m^2\right)\)
- For \(m^1\): \(\left(\frac{1}{3} m - \frac{1}{9} m\right)\)
Rearranging the expression by these groups, we get:
\[ -5 m^3 + \left(\frac{8}{9} + \frac{1}{9}\right) m^2 + \left(\frac{1}{3} - \frac{1}{9}\right) m \]
- Simplifying the coefficients
Now, we perform the arithmetic for the coefficients of each term:
- The coefficient of \(m^3\) is simply \(-5\).
- For \(m^2\), we have \(\frac{8}{9} + \frac{1}{9} = \frac{9}{9} = 1\), so the term becomes \(1 \cdot m^2 = m^2\).
- For \(m\), we find a common denominator of 9: \(\frac{1}{3} = \frac{3}{9}\). So, \(\frac{3}{9} - \frac{1}{9} = \frac{2}{9}\), and the term becomes \(\frac{2}{9} m\).
Combining these, the simplified polynomial is:
\[ -5 m^3 + m^2 + \frac{2}{9} m \]
Examples
Polynomial subtraction is a foundational skill in algebra, useful in scenarios like calculating net profit or loss. If a company's revenue is modeled by one polynomial \(P_R(m) = \frac{8}{9} m^2 + \frac{1}{3} m\) and its total costs are represented by \(P_C(m) = 5 m^3 - \frac{1}{9} m^2 + \frac{1}{9} m\) (where \(m\) could represent the number of units produced in thousands), finding the profit involves subtracting the cost from the revenue. Performing this subtraction gives the profit function, helping the business predict financial outcomes based on different production levels.