Answer :
To solve this problem, we need to perform polynomial long division to find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex].
### Step-by-Step Polynomial Long Division:
1. Initial Setup:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Division Process:
- First Term:
- Divide the first term of the dividend by the first term of the divisor: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex]: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x^2 + 0x - 15
\][/tex]
- Second Term:
- Now, divide the new first term of the remainder [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the entire divisor by [tex]\(5\)[/tex]: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this result from the current remainder:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex], the division results in a quotient of:
[tex]\[ x + 5 \][/tex]
3. Conclusion:
- The quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Therefore, the correct answer from the given options is [tex]\(x + 5\)[/tex].
### Step-by-Step Polynomial Long Division:
1. Initial Setup:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Division Process:
- First Term:
- Divide the first term of the dividend by the first term of the divisor: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex]: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x^2 + 0x - 15
\][/tex]
- Second Term:
- Now, divide the new first term of the remainder [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the entire divisor by [tex]\(5\)[/tex]: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this result from the current remainder:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex], the division results in a quotient of:
[tex]\[ x + 5 \][/tex]
3. Conclusion:
- The quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Therefore, the correct answer from the given options is [tex]\(x + 5\)[/tex].
