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Answer :
Let's solve the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex] using polynomial long division.
### Step-by-step Division:
1. 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].
- [tex]\(x^4 \div x^3 = x\)[/tex].
2. Multiply the entire divisor by this result:
- [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
3. Subtract to find the remainder:
- Subtract [tex]\((x^4 - 3x)\)[/tex] from [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex].
- [tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex].
5. Multiply and subtract:
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex].
- [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract:
- [tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Compilation of the Result:
- The quotient from this division is [tex]\(x + 5\)[/tex].
- The remainder is 0, confirming the division is exact.
So, the quotient of [tex]\(\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}\)[/tex] is [tex]\(x + 5\)[/tex].
The correct answer from the options given is [tex]\((x + 5)\)[/tex].
### Step-by-step Division:
1. 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].
- [tex]\(x^4 \div x^3 = x\)[/tex].
2. Multiply the entire divisor by this result:
- [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
3. Subtract to find the remainder:
- Subtract [tex]\((x^4 - 3x)\)[/tex] from [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex].
- [tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the process:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex].
5. Multiply and subtract:
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex].
- [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract:
- [tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
### Compilation of the Result:
- The quotient from this division is [tex]\(x + 5\)[/tex].
- The remainder is 0, confirming the division is exact.
So, the quotient of [tex]\(\frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3}\)[/tex] is [tex]\(x + 5\)[/tex].
The correct answer from the options given is [tex]\((x + 5)\)[/tex].
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