Answer :
- Factor the numerator by grouping: $x^4+5x^3-3x-15 = (x+5)(x^3-3)$.
- Divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
- Cancel the common factor $(x^3-3)$.
- The quotient is $x+5$, so 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 it is a polynomial.
2. Factoring the Numerator
We can try to factor the numerator to see if $x^3-3$ is a factor. We can use factoring by grouping:
$x^4+5x^3-3x-15 = x(x^3-3) + 5(x^3-3)$
3. Factoring by Grouping
Now we can factor out $(x^3-3)$ from both terms:
$x(x^3-3) + 5(x^3-3) = (x+5)(x^3-3)$
4. Rewriting the Expression
So we have $\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3}$.
5. Simplifying the Expression
Since $x^3-3$ is a factor of both the numerator and the denominator, we can cancel it out, provided that $x^3-3 \neq 0$:
$\frac{(x+5)(x^3-3)}{x^3-3} = x+5$
6. Final Answer
Therefore, the quotient is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and computer graphics. For example, when designing a digital filter, engineers use polynomial division to simplify the transfer function of the filter, making it easier to analyze and implement. Also, polynomial division can be used to determine the stability of a system or to find the roots of a polynomial, which can be useful in solving differential equations or modeling physical phenomena.
- Divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
- Cancel the common factor $(x^3-3)$.
- The quotient is $x+5$, so 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 it is a polynomial.
2. Factoring the Numerator
We can try to factor the numerator to see if $x^3-3$ is a factor. We can use factoring by grouping:
$x^4+5x^3-3x-15 = x(x^3-3) + 5(x^3-3)$
3. Factoring by Grouping
Now we can factor out $(x^3-3)$ from both terms:
$x(x^3-3) + 5(x^3-3) = (x+5)(x^3-3)$
4. Rewriting the Expression
So we have $\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3}$.
5. Simplifying the Expression
Since $x^3-3$ is a factor of both the numerator and the denominator, we can cancel it out, provided that $x^3-3 \neq 0$:
$\frac{(x+5)(x^3-3)}{x^3-3} = x+5$
6. Final Answer
Therefore, the quotient is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and computer graphics. For example, when designing a digital filter, engineers use polynomial division to simplify the transfer function of the filter, making it easier to analyze and implement. Also, polynomial division can be used to determine the stability of a system or to find the roots of a polynomial, which can be useful in solving differential equations or modeling physical phenomena.
