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Answer :
To find the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we can perform polynomial long division. Here’s how you can solve it step-by-step:
1. Set up the division: Place the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the long division symbol and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. Divide the leading term: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]. This gives you [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex]. Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat until done: Since the degree of the remainder [tex]\(5x^3 - 15\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], the current division gives the final quotient.
5. Conclusion: The quotient is [tex]\(x + 5\)[/tex] because the polynomial long division results in this expression as the simplified division result.
Therefore, the quotient of [tex]\((x^4 + 5 x^3 - 3 x - 15) \div (x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the division: Place the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the long division symbol and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. Divide the leading term: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]. This gives you [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex]. Subtract this result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat until done: Since the degree of the remainder [tex]\(5x^3 - 15\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], the current division gives the final quotient.
5. Conclusion: The quotient is [tex]\(x + 5\)[/tex] because the polynomial long division results in this expression as the simplified division result.
Therefore, the quotient of [tex]\((x^4 + 5 x^3 - 3 x - 15) \div (x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
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