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 we need to find the quotient, assuming 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 the expression:
$x(x^3-3) + 5(x^3-3) = (x+5)(x^3-3)$
4. Rewriting the Expression
Now we can rewrite the original expression as:
$\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3}$
5. Simplifying the Expression
Since $x^3-3$ is a factor of both the numerator and the denominator, we can cancel it out:
$\frac{(x+5)(x^3-3)}{x^3-3} = x+5$
6. Final Answer
Therefore, the quotient is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific fields. For example, when designing a bridge, engineers use polynomial functions to model the load distribution. Dividing one polynomial by another can help determine the stress at different points on the structure, ensuring the bridge's stability and safety. Similarly, in signal processing, polynomial division is used to analyze and filter signals, separating desired information from noise.
- 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 we need to find the quotient, assuming 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 the expression:
$x(x^3-3) + 5(x^3-3) = (x+5)(x^3-3)$
4. Rewriting the Expression
Now we can rewrite the original expression as:
$\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x+5)(x^3-3)}{x^3-3}$
5. Simplifying the Expression
Since $x^3-3$ is a factor of both the numerator and the denominator, we can cancel it out:
$\frac{(x+5)(x^3-3)}{x^3-3} = x+5$
6. Final Answer
Therefore, the quotient is $x+5$.
### Examples
Polynomial division is used in various engineering and scientific fields. For example, when designing a bridge, engineers use polynomial functions to model the load distribution. Dividing one polynomial by another can help determine the stress at different points on the structure, ensuring the bridge's stability and safety. Similarly, in signal processing, polynomial division is used to analyze and filter signals, separating desired information from noise.
