Answer :
To solve this problem, we need to perform polynomial long division to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
### Steps for Polynomial Long Division
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of our quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this quotient term:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
4. Repeat the process with the new polynomial:
- Divide the first term of the new result by the first term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the second term of our quotient is [tex]\(5\)[/tex].
5. Multiply the entire divisor by this quotient term:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we have reached a zero polynomial, and there are no remaining terms of higher degree than [tex]\(x^3\)[/tex], we can stop the division.
### Conclusion
The quotient from the division is [tex]\(x + 5\)[/tex].
Thus, the answer to the problem is:
[tex]\(x + 5\)[/tex]
### Steps for Polynomial Long Division
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of our quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this quotient term:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
4. Repeat the process with the new polynomial:
- Divide the first term of the new result by the first term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the second term of our quotient is [tex]\(5\)[/tex].
5. Multiply the entire divisor by this quotient term:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we have reached a zero polynomial, and there are no remaining terms of higher degree than [tex]\(x^3\)[/tex], we can stop the division.
### Conclusion
The quotient from the division is [tex]\(x + 5\)[/tex].
Thus, the answer to the problem is:
[tex]\(x + 5\)[/tex]
