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Answer :
- Expand the product of the polynomials $(x+2)$ and $(x^2-x-8)$: $(x+2)(x^2-x-8) = x^3 + x^2 - 10x - 16$.
- Multiply the resulting polynomial by 3: $3(x^3 + x^2 - 10x - 16) = 3x^3 + 3x^2 - 30x - 48$.
- The simplified expression is $3x^3 + 3x^2 - 30x - 48$.
- The final answer is $\boxed{3 x^3+3 x^2-30 x-48}$.
### Explanation
1. Understanding the Problem
We are given the expression $3(x+2)(x^2-x-8)$ and asked to simplify it. This involves expanding the product of a constant and two polynomials to obtain a simplified polynomial expression in standard form.
2. Expanding the Polynomials
First, let's expand the product of the two polynomials $(x+2)$ and $(x^2-x-8)$. We use the distributive property:
$$(x+2)(x^2-x-8) = x(x^2-x-8) + 2(x^2-x-8)$$
$$= x^3 - x^2 - 8x + 2x^2 - 2x - 16$$
Now, combine like terms:
$$= x^3 + (-1+2)x^2 + (-8-2)x - 16$$
$$= x^3 + x^2 - 10x - 16$$
3. Multiplying by the Constant
Next, we multiply the resulting polynomial by 3:
$$3(x^3 + x^2 - 10x - 16) = 3x^3 + 3x^2 - 30x - 48$$
4. Final Answer
The simplified expression is $3x^3 + 3x^2 - 30x - 48$. Comparing this with the given options, we find that it matches the first option.
### Examples
Polynomial simplification is a fundamental skill in algebra and is used in various real-world applications. For instance, when designing a bridge, engineers use polynomials to model the load and stress distribution. Simplifying these polynomial expressions helps them to analyze the structural integrity and safety of the bridge. Similarly, in economics, polynomial functions are used to model cost, revenue, and profit. Simplifying these functions allows economists to make informed decisions about pricing and production levels.
- Multiply the resulting polynomial by 3: $3(x^3 + x^2 - 10x - 16) = 3x^3 + 3x^2 - 30x - 48$.
- The simplified expression is $3x^3 + 3x^2 - 30x - 48$.
- The final answer is $\boxed{3 x^3+3 x^2-30 x-48}$.
### Explanation
1. Understanding the Problem
We are given the expression $3(x+2)(x^2-x-8)$ and asked to simplify it. This involves expanding the product of a constant and two polynomials to obtain a simplified polynomial expression in standard form.
2. Expanding the Polynomials
First, let's expand the product of the two polynomials $(x+2)$ and $(x^2-x-8)$. We use the distributive property:
$$(x+2)(x^2-x-8) = x(x^2-x-8) + 2(x^2-x-8)$$
$$= x^3 - x^2 - 8x + 2x^2 - 2x - 16$$
Now, combine like terms:
$$= x^3 + (-1+2)x^2 + (-8-2)x - 16$$
$$= x^3 + x^2 - 10x - 16$$
3. Multiplying by the Constant
Next, we multiply the resulting polynomial by 3:
$$3(x^3 + x^2 - 10x - 16) = 3x^3 + 3x^2 - 30x - 48$$
4. Final Answer
The simplified expression is $3x^3 + 3x^2 - 30x - 48$. Comparing this with the given options, we find that it matches the first option.
### Examples
Polynomial simplification is a fundamental skill in algebra and is used in various real-world applications. For instance, when designing a bridge, engineers use polynomials to model the load and stress distribution. Simplifying these polynomial expressions helps them to analyze the structural integrity and safety of the bridge. Similarly, in economics, polynomial functions are used to model cost, revenue, and profit. Simplifying these functions allows economists to make informed decisions about pricing and production levels.
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