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Answer :
- Distribute the negative sign: $(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$.
- Combine like terms: $5x^3 + (4x^2 - 6x^2) + 2x + 9$.
- Simplify: $5x^3 - 2x^2 + 2x + 9$.
- 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 the 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 find the difference, we distribute the negative sign to each term in the second polynomial:
$$(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$$
This step is crucial because it changes the signs of the terms being subtracted.
3. Combining Like Terms
Next, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $4x^2$ and $-6x^2$ as like terms. Combining them, we get:
$$5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9$$
So, $4x^2 - 6x^2 = -2x^2$.
4. Final Result
Therefore, the difference of the polynomials is $5x^3 - 2x^2 + 2x + 9$.
### Examples
Polynomial subtraction is used in various fields, such as engineering and computer graphics. For example, in computer graphics, polynomial subtraction can be used to calculate the difference between two curves or surfaces. In engineering, it can be used to analyze the behavior of systems described by polynomial equations. Understanding polynomial subtraction helps in modeling and solving real-world problems in these areas.
- Combine like terms: $5x^3 + (4x^2 - 6x^2) + 2x + 9$.
- Simplify: $5x^3 - 2x^2 + 2x + 9$.
- 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 the 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 find the difference, we distribute the negative sign to each term in the second polynomial:
$$(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$$
This step is crucial because it changes the signs of the terms being subtracted.
3. Combining Like Terms
Next, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have $4x^2$ and $-6x^2$ as like terms. Combining them, we get:
$$5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9$$
So, $4x^2 - 6x^2 = -2x^2$.
4. Final Result
Therefore, the difference of the polynomials is $5x^3 - 2x^2 + 2x + 9$.
### Examples
Polynomial subtraction is used in various fields, such as engineering and computer graphics. For example, in computer graphics, polynomial subtraction can be used to calculate the difference between two curves or surfaces. In engineering, it can be used to analyze the behavior of systems described by polynomial equations. Understanding polynomial subtraction helps in modeling and solving real-world problems in these areas.
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