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:
$$(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$$
3. Combining Like Terms
Now, 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 gives us:
$$5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9$$
4. Final Result
So, 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, subtracting polynomials can help determine the difference in light intensity between two objects, creating realistic shading effects. In engineering, it can be used to model the change in a system's behavior over time by subtracting one polynomial representing the initial state from another representing the final state.
- 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:
$$(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$$
3. Combining Like Terms
Now, 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 gives us:
$$5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9$$
4. Final Result
So, 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, subtracting polynomials can help determine the difference in light intensity between two objects, creating realistic shading effects. In engineering, it can be used to model the change in a system's behavior over time by subtracting one polynomial representing the initial state from another representing the final state.
