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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: $5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9$.
- 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:
$$(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$$
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:
$$4x^2 - 6x^2 = -2x^2$$
So the expression becomes:
$$5x^3 - 2x^2 + 2x + 9$$
4. Final Answer
Finally, we compare our result with the given options. The correct answer is:
$$5x^3 - 2x^2 + 2x + 9$$
### Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, computer graphics, and economics. For example, engineers use polynomials to design bridges and buildings, while economists use them to model economic growth. Understanding how to manipulate polynomials, such as finding their difference, is essential for solving real-world problems in these fields.
- Combine like terms: $5x^3 + (4x^2 - 6x^2) + 2x + 9 = 5x^3 - 2x^2 + 2x + 9$.
- 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:
$$(5x^3 + 4x^2) - (6x^2 - 2x - 9) = 5x^3 + 4x^2 - 6x^2 + 2x + 9$$
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:
$$4x^2 - 6x^2 = -2x^2$$
So the expression becomes:
$$5x^3 - 2x^2 + 2x + 9$$
4. Final Answer
Finally, we compare our result with the given options. The correct answer is:
$$5x^3 - 2x^2 + 2x + 9$$
### Examples
Polynomials are used to model curves and shapes in various fields, such as engineering, computer graphics, and economics. For example, engineers use polynomials to design bridges and buildings, while economists use them to model economic growth. Understanding how to manipulate polynomials, such as finding their difference, is essential for solving real-world problems in these fields.
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