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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], we can use polynomial long division.
Here’s how polynomial long division works:
1. Set up the division:
- The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor is [tex]\(x^3 - 3\)[/tex].
2. Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply the entire divisor by this term:
- Multiply [tex]\(x\)[/tex] by [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
4. Subtract this from the original dividend:
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15.
\][/tex]
5. Repeat the process:
- The new dividend is [tex]\(5x^3 + 3x - 15\)[/tex].
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
6. Multiply the entire divisor by this new term:
- Multiply [tex]\(5\)[/tex] by [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
7. Subtract this from the new dividend:
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 + 3x - 15\)[/tex]:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x.
\][/tex]
8. Since the degree of the remainder is less than the divisor, the division is complete.
The quotient of the division is [tex]\(x + 5\)[/tex], and there is no remainder ([tex]\(3x\)[/tex] minus 0 would have appeared as a remainder, but it's eliminated in this process).
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Here’s how polynomial long division works:
1. Set up the division:
- The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor is [tex]\(x^3 - 3\)[/tex].
2. Divide the first term of the dividend by the first term of the divisor:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply the entire divisor by this term:
- Multiply [tex]\(x\)[/tex] by [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
4. Subtract this from the original dividend:
- Subtract [tex]\(x^4 - 3x\)[/tex] from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15.
\][/tex]
5. Repeat the process:
- The new dividend is [tex]\(5x^3 + 3x - 15\)[/tex].
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
6. Multiply the entire divisor by this new term:
- Multiply [tex]\(5\)[/tex] by [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
7. Subtract this from the new dividend:
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 + 3x - 15\)[/tex]:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x.
\][/tex]
8. Since the degree of the remainder is less than the divisor, the division is complete.
The quotient of the division is [tex]\(x + 5\)[/tex], and there is no remainder ([tex]\(3x\)[/tex] minus 0 would have appeared as a remainder, but it's eliminated in this process).
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
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