Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Here's how it works:
1. Setup the Division: Place the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the division bar and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. First Division Step:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend, resulting in a new polynomial: [tex]\(5x^3 - 3x - 15 - (x^4 - 3x) = 5x^3 - 15\)[/tex].
3. Second Division Step:
- Divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(+5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract to update the remainder: [tex]\(5x^3 - 15 - (5x^3 - 15) = 0\)[/tex].
Since our remainder is 0, the division ends here.
Thus, the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Therefore, the correct quotient is [tex]\(x + 5\)[/tex].
1. Setup the Division: Place the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] under the division bar and the divisor [tex]\(x^3 - 3\)[/tex] outside.
2. First Division Step:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend, resulting in a new polynomial: [tex]\(5x^3 - 3x - 15 - (x^4 - 3x) = 5x^3 - 15\)[/tex].
3. Second Division Step:
- Divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(+5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract to update the remainder: [tex]\(5x^3 - 15 - (5x^3 - 15) = 0\)[/tex].
Since our remainder is 0, the division ends here.
Thus, the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Therefore, the correct quotient is [tex]\(x + 5\)[/tex].
