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Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Here's how it works step-by-step:
1. Set Up the Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. First Division Step:
- Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]):
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0 - 15
\][/tex]
3. Second Division Step:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract this result from the remaining dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
4. Conclusion:
- The remainder is 0, which means [tex]\(x\)[/tex] from the first step and [tex]\(5\)[/tex] from the second step can be added together to form the quotient:
[tex]\[
\text{Quotient} = x + 5
\][/tex]
Hence, the quotient of the polynomial division is [tex]\(x + 5\)[/tex]. Therefore, the correct choice from the given options is [tex]\(x+5\)[/tex].
1. Set Up the Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. First Division Step:
- Divide the leading term of the dividend ([tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]):
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0 - 15
\][/tex]
3. Second Division Step:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract this result from the remaining dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
4. Conclusion:
- The remainder is 0, which means [tex]\(x\)[/tex] from the first step and [tex]\(5\)[/tex] from the second step can be added together to form the quotient:
[tex]\[
\text{Quotient} = x + 5
\][/tex]
Hence, the quotient of the polynomial division is [tex]\(x + 5\)[/tex]. Therefore, the correct choice from the given options is [tex]\(x+5\)[/tex].
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