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Answer :
To find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Let's go through this process step-by-step:
1. Set up the division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first term of the dividend by the first term of the divisor:
- [tex]\(\frac{x^4}{x^3} = x\)[/tex]
3. Multiply the entire divisor by this result:
- [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex]
4. Subtract the result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
5. Divide the first term of the new polynomial by the first term of the divisor again:
- [tex]\(\frac{5x^3}{x^3} = 5\)[/tex]
6. Multiply the entire divisor by this result:
- [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex]
7. Subtract this from the previous result:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
8. Now, since all terms in the new polynomial are of lower degree than the divisor, the division process stops, and the remainder is simply what's left:
- Remainder: [tex]\(3x\)[/tex]
The quotient is what we obtained through our division steps, which is [tex]\(x + 5\)[/tex].
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] 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. Divide the first term of the dividend by the first term of the divisor:
- [tex]\(\frac{x^4}{x^3} = x\)[/tex]
3. Multiply the entire divisor by this result:
- [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex]
4. Subtract the result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 3x - 15
\][/tex]
5. Divide the first term of the new polynomial by the first term of the divisor again:
- [tex]\(\frac{5x^3}{x^3} = 5\)[/tex]
6. Multiply the entire divisor by this result:
- [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex]
7. Subtract this from the previous result:
[tex]\[
(5x^3 + 3x - 15) - (5x^3 - 15) = 3x
\][/tex]
8. Now, since all terms in the new polynomial are of lower degree than the divisor, the division process stops, and the remainder is simply what's left:
- Remainder: [tex]\(3x\)[/tex]
The quotient is what we obtained through our division steps, which is [tex]\(x + 5\)[/tex].
Thus, 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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