Answer :
To find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division.
1. Set Up the Division: Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the division symbol, and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(x^4 \div x^3 = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
4. Bring Down the Next Term: There’s no x-squared term to bring down, so we deal directly with [tex]\(5x^3\)[/tex].
5. Repeat the Process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex].
6. Multiply and Subtract Again:
- Multiply the divisor by [tex]\(5\)[/tex], giving [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this from the current dividend remainder:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since there is no remainder, the division process ends here.
Conclusion: The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
So, the correct answer from the options is [tex]\(x + 5\)[/tex].
1. Set Up the Division: Write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the division symbol, and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- [tex]\(x^4 \div x^3 = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], giving [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
4. Bring Down the Next Term: There’s no x-squared term to bring down, so we deal directly with [tex]\(5x^3\)[/tex].
5. Repeat the Process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex].
6. Multiply and Subtract Again:
- Multiply the divisor by [tex]\(5\)[/tex], giving [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this from the current dividend remainder:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since there is no remainder, the division process ends here.
Conclusion: The quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
So, the correct answer from the options is [tex]\(x + 5\)[/tex].
