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 $\boxed{x+5}$.
### Explanation
1. Understanding the Problem
We are given the expression $\frac{x^4+5x^3-3x-15}{x^3-3}$ and asked to find the quotient, assuming it is a polynomial. We can solve this by polynomial long division or by factoring the numerator.
2. Factoring the Numerator
Let's try factoring by grouping. We can rewrite the numerator as follows:
$$x^4+5x^3-3x-15 = x(x^3-3) + 5(x^3-3)$$
$$= (x+5)(x^3-3)$$
3. Simplifying the Expression
Now we can simplify the expression:
$$\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3}$$
$$= x+5$$
4. Final Answer
Therefore, the quotient is $x+5$.
### Examples
Polynomial division is used in various engineering fields, such as control systems, to analyze the stability and response of systems. For example, when designing a feedback control system, engineers often use polynomial division to simplify transfer functions and determine the system's behavior. This helps them ensure that the system operates correctly and remains stable under different conditions. Understanding polynomial division allows engineers to optimize system performance and prevent unwanted oscillations or instability.
- 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 $\boxed{x+5}$.
### Explanation
1. Understanding the Problem
We are given the expression $\frac{x^4+5x^3-3x-15}{x^3-3}$ and asked to find the quotient, assuming it is a polynomial. We can solve this by polynomial long division or by factoring the numerator.
2. Factoring the Numerator
Let's try factoring by grouping. We can rewrite the numerator as follows:
$$x^4+5x^3-3x-15 = x(x^3-3) + 5(x^3-3)$$
$$= (x+5)(x^3-3)$$
3. Simplifying the Expression
Now we can simplify the expression:
$$\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3}$$
$$= x+5$$
4. Final Answer
Therefore, the quotient is $x+5$.
### Examples
Polynomial division is used in various engineering fields, such as control systems, to analyze the stability and response of systems. For example, when designing a feedback control system, engineers often use polynomial division to simplify transfer functions and determine the system's behavior. This helps them ensure that the system operates correctly and remains stable under different conditions. Understanding polynomial division allows engineers to optimize system performance and prevent unwanted oscillations or instability.
