Answer :
- Factor the numerator by grouping: $x^4+5x^3-3x-15 = (x^3-3)(x+5)$.
- Rewrite the expression as $\frac{(x^3-3)(x+5)}{x^3-3}$.
- 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 want 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: $$x^4+5x^3-3x-15 = x^3(x+5) - 3(x+5) = (x^3-3)(x+5)$$.
3. Rewriting the Expression
Now we can rewrite the expression as: $$\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x^3-3)(x+5)}{x^3-3}$$.
4. Simplifying the Expression
Since $x^3-3$ is a factor of the numerator, we can cancel it out, which gives us: $$\frac{(x^3-3)(x+5)}{x^3-3} = x+5$$.
5. Final Answer
Therefore, the quotient 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 example, when designing a bridge, engineers use polynomial functions to model the load distribution and structural integrity. Dividing these polynomials helps them determine critical parameters and ensure the bridge's stability under different conditions.
- Rewrite the expression as $\frac{(x^3-3)(x+5)}{x^3-3}$.
- 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 want 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: $$x^4+5x^3-3x-15 = x^3(x+5) - 3(x+5) = (x^3-3)(x+5)$$.
3. Rewriting the Expression
Now we can rewrite the expression as: $$\frac{x^4+5x^3-3x-15}{x^3-3} = \frac{(x^3-3)(x+5)}{x^3-3}$$.
4. Simplifying the Expression
Since $x^3-3$ is a factor of the numerator, we can cancel it out, which gives us: $$\frac{(x^3-3)(x+5)}{x^3-3} = x+5$$.
5. Final Answer
Therefore, the quotient 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 example, when designing a bridge, engineers use polynomial functions to model the load distribution and structural integrity. Dividing these polynomials helps them determine critical parameters and ensure the bridge's stability under different conditions.
