- Distribute the negative sign to each term of the second polynomial: \(-(6x^2 - 2x - 9) = -6x^2 + 2x + 9\).
- Combine the result with the first polynomial: \(5x^3 + 4x^2 - 6x^2 + 2x + 9\).
- Simplify by grouping like terms: \(5x^3 + (4-6)x^2 + 2x + 9\).
- Final expression: \(\boxed{5x^3 - 2x^2 + 2x + 9}\).
Explanation
- Analyzing the Problem
The objective is to compute the difference between two given polynomials: \((5x^3 + 4x^2)\) and \((6x^2 - 2x - 9)\). Polynomial subtraction involves distributing the negative sign throughout the second polynomial and then combining like terms. Like terms are terms that have the same variable raised to the same power. Here, we identify the terms \(x^3\), \(x^2\), \(x^1\), and constant terms.
- Distributing the Negative Sign
To subtract the polynomials, we distribute the negative sign to each term inside the second parentheses:
\[-(6x^2 - 2x - 9) = -6x^2 + 2x + 9.\]
Now, rewrite the full expression by combining this with the first polynomial:
\[(5x^3 + 4x^2) - 6x^2 + 2x + 9.\]
- Combining Like Terms
Next, we group the terms with the same powers of \(x\):
\[5x^3 + (4x^2 - 6x^2) + 2x + 9.\]
Now, simplify the coefficients for the \(x^2\) term:
\[4x^2 - 6x^2 = (4 - 6)x^2 = -2x^2.\]
Substituting this back into the expression, we get:
\[5x^3 - 2x^2 + 2x + 9.\]
- Final Conclusion
Comparing our result \(5x^3 - 2x^2 + 2x + 9\) with the given options:
- \(-x^3+6 x^2+9\)
- \(-x^3+2 x^2-9\)
- \(5 x^3-2 x^2-2 x-9\)
- \(5 x^3-2 x^2+2 x+9\)
Our result matches the fourth option. Therefore, the difference of the polynomials is \(\boxed{5x^3 - 2x^2 + 2x + 9}\).
Examples
Polynomial subtraction is fundamental in algebra for simplifying expressions, such as calculating the difference in profit or growth over time. For example, if a company's revenue is modeled by \(R(x) = 5x^3 + 4x^2\) and their costs are modeled by \(C(x) = 6x^2 - 2x - 9\), the profit function \(P(x) = R(x) - C(x)\) is found by subtracting these polynomials, allowing analysts to determine the net earnings based on variable \(x\). This helps in making informed financial decisions.