Answer :
To solve the problem of finding the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex], we go through polynomial division.
### Step-by-Step Solution:
1. Set Up the Division: Write the dividend [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the division bar and the divisor [tex]\((x^3 - 3)\)[/tex] on the outside.
2. Divide the Leading Terms:
- Look at the leading terms of the dividend and the divisor.
- Dividend’s leading term: [tex]\(x^4\)[/tex]
- Divisor’s leading term: [tex]\(x^3\)[/tex]
- Divide these leading terms to find the first term of the quotient: [tex]\((x^4) \div (x^3) = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient [tex]\(x\)[/tex].
- This results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\((x^4 - 3x)\)[/tex] 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:
- Use [tex]\(5x^3\)[/tex] and [tex]\(x^3\)[/tex] leading terms: [tex]\((5x^3) \div (x^3) = 5\)[/tex].
- Multiply the entire divisor by this new term of the quotient:
[tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from what we have or the previous result:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0x^2 + 0x + 0
\][/tex]
Since there are no more terms left to bring down, and the degree of the remaining polynomial under the division bar is less than that of the divisor, the division is complete.
### Conclusion:
The quotient of the division is [tex]\(x + 5\)[/tex], and the remainder is 0. Thus, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is simply [tex]\((x + 5)\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{x + 5}\)[/tex].
### Step-by-Step Solution:
1. Set Up the Division: Write the dividend [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the division bar and the divisor [tex]\((x^3 - 3)\)[/tex] on the outside.
2. Divide the Leading Terms:
- Look at the leading terms of the dividend and the divisor.
- Dividend’s leading term: [tex]\(x^4\)[/tex]
- Divisor’s leading term: [tex]\(x^3\)[/tex]
- Divide these leading terms to find the first term of the quotient: [tex]\((x^4) \div (x^3) = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by the first term of the quotient [tex]\(x\)[/tex].
- This results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\((x^4 - 3x)\)[/tex] 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:
- Use [tex]\(5x^3\)[/tex] and [tex]\(x^3\)[/tex] leading terms: [tex]\((5x^3) \div (x^3) = 5\)[/tex].
- Multiply the entire divisor by this new term of the quotient:
[tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from what we have or the previous result:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0x^2 + 0x + 0
\][/tex]
Since there are no more terms left to bring down, and the degree of the remaining polynomial under the division bar is less than that of the divisor, the division is complete.
### Conclusion:
The quotient of the division is [tex]\(x + 5\)[/tex], and the remainder is 0. Thus, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is simply [tex]\((x + 5)\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{x + 5}\)[/tex].
