Answer :
To solve the problem of finding the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], you can use polynomial long division. Here's a step-by-step explanation of how you would perform the division:
1. Set up the division: Write 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 you [tex]\(x\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(x\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex] which gives [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex]. This subtraction yields a new polynomial of [tex]\(5x^3 + 0x^2 - 15\)[/tex].
4. Repeat the process:
- Divide the leading term of the new dividend, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], which gives you [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the remaining polynomial [tex]\(5x^3 + 0x^2 - 15\)[/tex] to see if there is any remainder.
5. Recognize completion: After performing the division steps, the operations complete when the degree of the remaining polynomial (if there is any) is less than the degree of the divisor.
From the steps above, the quotient obtained is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the division: Write 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 you [tex]\(x\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(x\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex] which gives [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex]. This subtraction yields a new polynomial of [tex]\(5x^3 + 0x^2 - 15\)[/tex].
4. Repeat the process:
- Divide the leading term of the new dividend, [tex]\(5x^3\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex], which gives you [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the remaining polynomial [tex]\(5x^3 + 0x^2 - 15\)[/tex] to see if there is any remainder.
5. Recognize completion: After performing the division steps, the operations complete when the degree of the remaining polynomial (if there is any) is less than the degree of the divisor.
From the steps above, the quotient obtained is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
