Answer :
To find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division.
### Step-by-step Polynomial Division:
1. Divide the leading terms:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex].
- This gives us [tex]\(x\)[/tex] as the first term of the quotient.
2. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the term [tex]\(x\)[/tex] of the quotient. This results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15
\][/tex]
- Simplifying gives [tex]\(5x^3 - 15\)[/tex].
3. Repeat the process:
- Since the degree of the remaining polynomial [tex]\(5x^3 - 15\)[/tex] is not less than the degree of the divisor, we repeat the process.
- But notice, the reduction already completed the division because after the last step subtraction, the resulting terms of degree are lower than x^3.
Therefore, the quotient of the given polynomials is [tex]\((x + 5)\)[/tex].
Thus, the correct answer to the problem is [tex]\(x + 5\)[/tex].
### Step-by-step Polynomial Division:
1. Divide the leading terms:
- Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex].
- This gives us [tex]\(x\)[/tex] as the first term of the quotient.
2. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the term [tex]\(x\)[/tex] of the quotient. This results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15
\][/tex]
- Simplifying gives [tex]\(5x^3 - 15\)[/tex].
3. Repeat the process:
- Since the degree of the remaining polynomial [tex]\(5x^3 - 15\)[/tex] is not less than the degree of the divisor, we repeat the process.
- But notice, the reduction already completed the division because after the last step subtraction, the resulting terms of degree are lower than x^3.
Therefore, the quotient of the given polynomials is [tex]\((x + 5)\)[/tex].
Thus, the correct answer to the problem is [tex]\(x + 5\)[/tex].
