Answer :
To solve the problem of finding the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex], we can perform polynomial long division.
1. Setup the Division: We want to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [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 and Subtract:
- Multiply the whole 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 [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- After subtraction, the new polynomial is [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
4. Repeat the Process:
- Next, divide the leading term of the new polynomial, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- This gives [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
- Multiply the whole divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(+5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the current polynomial [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
- After subtraction, the remainder is [tex]\(0\)[/tex].
6. Conclusion:
- Once we get a remainder of 0, we have our quotient.
- So, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex].
Therefore, the quotient is [tex]\(x + 5\)[/tex].
1. Setup the Division: We want to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [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 and Subtract:
- Multiply the whole 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 [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- After subtraction, the new polynomial is [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
4. Repeat the Process:
- Next, divide the leading term of the new polynomial, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex].
- This gives [tex]\(+5\)[/tex].
5. Multiply and Subtract Again:
- Multiply the whole divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(+5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the current polynomial [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
- After subtraction, the remainder is [tex]\(0\)[/tex].
6. Conclusion:
- Once we get a remainder of 0, we have our quotient.
- So, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex].
Therefore, the quotient is [tex]\(x + 5\)[/tex].
