Answer :
To solve the problem of finding the quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we can use polynomial long division. Here's a detailed, step-by-step approach:
1. Setup the Division: The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor is [tex]\(x^3 - 3\)[/tex].
2. First Division Step:
- Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]), which gives [tex]\(x\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this result [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], which results in [tex]\(5x^3 + 0x^2 - 15\)[/tex].
3. Second Division Step:
- Now, take the leading term of the new dividend ([tex]\(5x^3\)[/tex]) and divide by the leading term of the divisor ([tex]\(x^3\)[/tex]), which gives [tex]\(5\)[/tex].
- Multiply the entire divisor by this result [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 + 0x^2 - 15\)[/tex], which leaves a remainder of [tex]\(0\)[/tex].
After these steps, the quotient of the division is [tex]\(x + 5\)[/tex], and the remainder is [tex]\(0\)[/tex]. This means the division results in a clean quotient with no remainder.
Therefore, the quotient of the polynomial division is [tex]\(\boxed{x + 5}\)[/tex].
1. Setup the Division: The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor is [tex]\(x^3 - 3\)[/tex].
2. First Division Step:
- Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]), which gives [tex]\(x\)[/tex].
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this result [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], which results in [tex]\(5x^3 + 0x^2 - 15\)[/tex].
3. Second Division Step:
- Now, take the leading term of the new dividend ([tex]\(5x^3\)[/tex]) and divide by the leading term of the divisor ([tex]\(x^3\)[/tex]), which gives [tex]\(5\)[/tex].
- Multiply the entire divisor by this result [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 + 0x^2 - 15\)[/tex], which leaves a remainder of [tex]\(0\)[/tex].
After these steps, the quotient of the division is [tex]\(x + 5\)[/tex], and the remainder is [tex]\(0\)[/tex]. This means the division results in a clean quotient with no remainder.
Therefore, the quotient of the polynomial division is [tex]\(\boxed{x + 5}\)[/tex].
