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Answer :
To solve the problem of finding the quotient when dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], you can follow the polynomial long division process:
1. Setup: Start by writing the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
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]. This gives [tex]\(x\)[/tex].
3. Multiply: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
4. Subtract: Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
5. Repeat: Since the remaining polynomial [tex]\(5x^3 - 15\)[/tex] cannot be divided further by [tex]\(x^3 - 3\)[/tex], observe the degree of the remainder (which is less than the degree of the divisor), concluding that [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] is exactly divided by [tex]\((x^3 - 3)\)[/tex] with a remainder of zero.
Thus, the quotient of the division is [tex]\((x + 5)\)[/tex] with no remainder, meaning [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] simplifies to the polynomial [tex]\((x + 5)\)[/tex].
1. Setup: Start by writing the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
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]. This gives [tex]\(x\)[/tex].
3. Multiply: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
4. Subtract: Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
5. Repeat: Since the remaining polynomial [tex]\(5x^3 - 15\)[/tex] cannot be divided further by [tex]\(x^3 - 3\)[/tex], observe the degree of the remainder (which is less than the degree of the divisor), concluding that [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] is exactly divided by [tex]\((x^3 - 3)\)[/tex] with a remainder of zero.
Thus, the quotient of the division is [tex]\((x + 5)\)[/tex] with no remainder, meaning [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] simplifies to the polynomial [tex]\((x + 5)\)[/tex].
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