Answer :
Sure! Let's find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] by performing polynomial long division.
### Steps for Polynomial Long Division:
1. Setup the Division:
[tex]\[
\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}
\][/tex]
2. First Division Step:
- Divide the first term of the numerator [tex]\(x^4\)[/tex] by the first term of the denominator [tex]\(x^3\)[/tex]:
[tex]\[
x^4 \div x^3 = x
\][/tex]
- Multiply [tex]\(x\)[/tex] by the entire denominator [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15 = 5x^3 - 15
\][/tex]
3. Second Division Step:
- Divide the first term of the new numerator [tex]\(5x^3\)[/tex] by the first term of the denominator [tex]\(x^3\)[/tex]:
[tex]\[
5x^3 \div x^3 = 5
\][/tex]
- Multiply [tex]\(5\)[/tex] by the entire denominator [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from the new numerator:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Conclusion:
The quotient is formed from the sums of these individual steps:
[tex]\[
x + 5
\][/tex]
### Final Result:
Therefore, the quotient of [tex]\(\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}\)[/tex] is:
[tex]\[
\boxed{x + 5}
\][/tex]
### Steps for Polynomial Long Division:
1. Setup the Division:
[tex]\[
\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}
\][/tex]
2. First Division Step:
- Divide the first term of the numerator [tex]\(x^4\)[/tex] by the first term of the denominator [tex]\(x^3\)[/tex]:
[tex]\[
x^4 \div x^3 = x
\][/tex]
- Multiply [tex]\(x\)[/tex] by the entire denominator [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15 = 5x^3 - 15
\][/tex]
3. Second Division Step:
- Divide the first term of the new numerator [tex]\(5x^3\)[/tex] by the first term of the denominator [tex]\(x^3\)[/tex]:
[tex]\[
5x^3 \div x^3 = 5
\][/tex]
- Multiply [tex]\(5\)[/tex] by the entire denominator [tex]\((x^3 - 3)\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract this from the new numerator:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Conclusion:
The quotient is formed from the sums of these individual steps:
[tex]\[
x + 5
\][/tex]
### Final Result:
Therefore, the quotient of [tex]\(\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}\)[/tex] is:
[tex]\[
\boxed{x + 5}
\][/tex]
