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Answer :
To find the quotient of the polynomial division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], follow these steps:
1. Setup the Division Problem:
- We have the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Perform Long Division:
- Step 1: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives us [tex]\(x\)[/tex].
- Step 2: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\((x^3 - 3)\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Step 3: Subtract this result from the original dividend. This cancellation leaves [tex]\(5x^3 - 3x + 3x - 15\)[/tex], simplifying to [tex]\(5x^3 - 15\)[/tex].
- Step 4: Repeat the process with the new dividend [tex]\(5x^3 - 15\)[/tex].
3. Continue Division:
- The process will show that the terms are further reduced similar to that step, and since the original problem determines the end results quicker:
- The quotient result simplifies through each efficient step to eventually point to: [tex]\(x + 5\)[/tex].
4. Conclusion:
- After careful polynomial division, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
So, the answer to the question is [tex]\(x + 5\)[/tex].
1. Setup the Division Problem:
- We have the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Perform Long Division:
- Step 1: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives us [tex]\(x\)[/tex].
- Step 2: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\((x^3 - 3)\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Step 3: Subtract this result from the original dividend. This cancellation leaves [tex]\(5x^3 - 3x + 3x - 15\)[/tex], simplifying to [tex]\(5x^3 - 15\)[/tex].
- Step 4: Repeat the process with the new dividend [tex]\(5x^3 - 15\)[/tex].
3. Continue Division:
- The process will show that the terms are further reduced similar to that step, and since the original problem determines the end results quicker:
- The quotient result simplifies through each efficient step to eventually point to: [tex]\(x + 5\)[/tex].
4. Conclusion:
- After careful polynomial division, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
So, the answer to the question is [tex]\(x + 5\)[/tex].
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