Answer :
Sure! Let's solve the problem step by step to find the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex].
1. Identify the dividend and the divisor:
- The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor is [tex]\(x^3 - 3\)[/tex].
2. Perform polynomial division:
- We divide the leading term of the dividend by the leading term of the divisor:
- The leading term of the dividend is [tex]\(x^4\)[/tex].
- The leading term of the divisor is [tex]\(x^3\)[/tex].
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by the quotient term [tex]\(x\)[/tex], which gives [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
- The result is [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
4. Repeat the process:
- Since the degree of the new dividend (now [tex]\(5x^3\)[/tex]) is equal to the degree of the divisor, repeat the division process.
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(+5\)[/tex].
- Multiply and subtract again:
[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5(x^3 - 3)) = 5x^3 - 15
\][/tex]
[tex]\[
5x^3 - 15 - 5x^3 + 15 = 0
\][/tex]
5. Conclusion:
- Once we obtain a remainder of 0, we know that the division is complete.
- Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
Therefore, the polynomial quotient is [tex]\(\boxed{x + 5}\)[/tex].
1. Identify the dividend and the divisor:
- The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor is [tex]\(x^3 - 3\)[/tex].
2. Perform polynomial division:
- We divide the leading term of the dividend by the leading term of the divisor:
- The leading term of the dividend is [tex]\(x^4\)[/tex].
- The leading term of the divisor is [tex]\(x^3\)[/tex].
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by the quotient term [tex]\(x\)[/tex], which gives [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
- The result is [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
4. Repeat the process:
- Since the degree of the new dividend (now [tex]\(5x^3\)[/tex]) is equal to the degree of the divisor, repeat the division process.
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(+5\)[/tex].
- Multiply and subtract again:
[tex]\[
(5x^3 + 0x^2 - 0x - 15) - (5(x^3 - 3)) = 5x^3 - 15
\][/tex]
[tex]\[
5x^3 - 15 - 5x^3 + 15 = 0
\][/tex]
5. Conclusion:
- Once we obtain a remainder of 0, we know that the division is complete.
- Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\((x + 5)\)[/tex].
Therefore, the polynomial quotient is [tex]\(\boxed{x + 5}\)[/tex].
