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Answer :
- Distribute the negative sign: $-(6x^2 - 2x - 9) = -6x^2 + 2x + 9$.
- Combine like terms: $(5x^3 + 4x^2) + (-6x^2 + 2x + 9) = 5x^3 + (4x^2 - 6x^2) + 2x + 9$.
- Simplify the expression: $5x^3 - 2x^2 + 2x + 9$.
- The difference of the polynomials is $\boxed{5 x^3-2 x^2+2 x+9}$.
### Explanation
1. Understanding the Problem
We are asked to find the difference between two polynomials: $(5x^3 + 4x^2)$ and $(6x^2 - 2x - 9)$. This means we need to subtract the second polynomial from the first.
2. Distributing the Negative Sign
To subtract the polynomials, we distribute the negative sign to each term in the second polynomial: $-(6x^2 - 2x - 9) = -6x^2 + 2x + 9$.
3. Combining Like Terms
Now, we combine like terms by adding the result to the first polynomial: $(5x^3 + 4x^2) + (-6x^2 + 2x + 9)$. This simplifies to $5x^3 + (4x^2 - 6x^2) + 2x + 9$.
4. Simplifying the Expression
Finally, we simplify the expression: $5x^3 - 2x^2 + 2x + 9$.
5. Final Answer
The difference of the polynomials $(5x^3 + 4x^2) - (6x^2 - 2x - 9)$ is $5x^3 - 2x^2 + 2x + 9$.
### Examples
Polynomial subtraction is used in various fields, such as engineering, computer graphics, and economics. For example, in engineering, it can be used to determine the net force acting on an object when multiple forces are applied. In computer graphics, it can be used to manipulate curves and surfaces. In economics, it can be used to model the difference between revenue and cost functions to determine profit.
- Combine like terms: $(5x^3 + 4x^2) + (-6x^2 + 2x + 9) = 5x^3 + (4x^2 - 6x^2) + 2x + 9$.
- Simplify the expression: $5x^3 - 2x^2 + 2x + 9$.
- The difference of the polynomials is $\boxed{5 x^3-2 x^2+2 x+9}$.
### Explanation
1. Understanding the Problem
We are asked to find the difference between two polynomials: $(5x^3 + 4x^2)$ and $(6x^2 - 2x - 9)$. This means we need to subtract the second polynomial from the first.
2. Distributing the Negative Sign
To subtract the polynomials, we distribute the negative sign to each term in the second polynomial: $-(6x^2 - 2x - 9) = -6x^2 + 2x + 9$.
3. Combining Like Terms
Now, we combine like terms by adding the result to the first polynomial: $(5x^3 + 4x^2) + (-6x^2 + 2x + 9)$. This simplifies to $5x^3 + (4x^2 - 6x^2) + 2x + 9$.
4. Simplifying the Expression
Finally, we simplify the expression: $5x^3 - 2x^2 + 2x + 9$.
5. Final Answer
The difference of the polynomials $(5x^3 + 4x^2) - (6x^2 - 2x - 9)$ is $5x^3 - 2x^2 + 2x + 9$.
### Examples
Polynomial subtraction is used in various fields, such as engineering, computer graphics, and economics. For example, in engineering, it can be used to determine the net force acting on an object when multiple forces are applied. In computer graphics, it can be used to manipulate curves and surfaces. In economics, it can be used to model the difference between revenue and cost functions to determine profit.
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