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Answer :
- Factor the numerator by grouping: $x^4+5x^3-3x-15 = x^3(x+5) - 3(x+5)$.
- Factor out the common term: $x^3(x+5) - 3(x+5) = (x+5)(x^3-3)$.
- Divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
- Cancel the common factor: $\frac{(x+5)(x^3-3)}{x^3-3} = x+5$. The final answer is $\boxed{x+5}$
### Explanation
1. Understanding the Problem
We are given the expression $\frac{x^4+5x^3-3x-15}{x^3-3}$ and we want to find the quotient, given that the quotient is a polynomial. We can factor the numerator by grouping.
2. Factoring by Grouping
We can rewrite the numerator as $x^3(x+5) - 3(x+5)$.
3. Factoring out Common Term
Now we factor out the common term $(x+5)$ to get $(x+5)(x^3-3)$.
4. Dividing by Denominator
Now we divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
5. Canceling Common Factor
We can cancel the common factor $(x^3-3)$ from the numerator and denominator.
6. Final Answer
The result is the quotient $x+5$. Therefore, the quotient of $\frac{x^4+5x^3-3x-15}{x^3-3}$ is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as designing control systems, analyzing circuits, and modeling physical phenomena. For instance, when designing a filter for a signal, engineers use polynomial division to simplify the transfer function, making it easier to analyze and implement the filter.
- Factor out the common term: $x^3(x+5) - 3(x+5) = (x+5)(x^3-3)$.
- Divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
- Cancel the common factor: $\frac{(x+5)(x^3-3)}{x^3-3} = x+5$. The final answer is $\boxed{x+5}$
### Explanation
1. Understanding the Problem
We are given the expression $\frac{x^4+5x^3-3x-15}{x^3-3}$ and we want to find the quotient, given that the quotient is a polynomial. We can factor the numerator by grouping.
2. Factoring by Grouping
We can rewrite the numerator as $x^3(x+5) - 3(x+5)$.
3. Factoring out Common Term
Now we factor out the common term $(x+5)$ to get $(x+5)(x^3-3)$.
4. Dividing by Denominator
Now we divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
5. Canceling Common Factor
We can cancel the common factor $(x^3-3)$ from the numerator and denominator.
6. Final Answer
The result is the quotient $x+5$. Therefore, the quotient of $\frac{x^4+5x^3-3x-15}{x^3-3}$ is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as designing control systems, analyzing circuits, and modeling physical phenomena. For instance, when designing a filter for a signal, engineers use polynomial division to simplify the transfer function, making it easier to analyze and implement the filter.
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