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Answer :
- Factor the numerator by grouping: $x^4+5x^3-3x-15 = (x+5)(x^3-3)$.
- Divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
- Simplify the expression by canceling out 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 the quotient is a polynomial. We can try to factor the numerator to see if $x^3-3$ is a factor.
2. Factoring the Numerator
We can factor the numerator by grouping: $x^4+5x^3-3x-15 = x(x^3-3) + 5(x^3-3) = (x+5)(x^3-3)$.
3. Simplifying the Expression
Therefore, $\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3} = x+5$.
4. Finding the Quotient
The quotient is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and data analysis. For example, engineers use polynomial division to simplify transfer functions in control systems, making it easier to analyze and design controllers. In signal processing, polynomial division helps in filter design and signal reconstruction.
- Divide the factored numerator by the denominator: $\frac{(x+5)(x^3-3)}{x^3-3}$.
- Simplify the expression by canceling out 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 the quotient is a polynomial. We can try to factor the numerator to see if $x^3-3$ is a factor.
2. Factoring the Numerator
We can factor the numerator by grouping: $x^4+5x^3-3x-15 = x(x^3-3) + 5(x^3-3) = (x+5)(x^3-3)$.
3. Simplifying the Expression
Therefore, $\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3} = x+5$.
4. Finding the Quotient
The quotient is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific applications, such as control systems design, signal processing, and data analysis. For example, engineers use polynomial division to simplify transfer functions in control systems, making it easier to analyze and design controllers. In signal processing, polynomial division helps in filter design and signal reconstruction.
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