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Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial long division. Here's how the division process works:
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], which gives [tex]\(x\)[/tex].
2. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from step 1, [tex]\(x\)[/tex].
- [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this product from the original polynomial:
- [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15\)[/tex].
3. Repeat the Process:
- Now, take the new polynomial [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(+5\)[/tex].
- Multiply the entire divisor by this quotient term:
- [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this product from the current polynomial:
- [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
Since the remainder is zero, and all terms have been divided, the division process ends here. The quotient is [tex]\(x + 5\)[/tex], which is a polynomial.
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is the polynomial [tex]\(x + 5\)[/tex].
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], which gives [tex]\(x\)[/tex].
2. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from step 1, [tex]\(x\)[/tex].
- [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this product from the original polynomial:
- [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15\)[/tex].
3. Repeat the Process:
- Now, take the new polynomial [tex]\(5x^3 - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(+5\)[/tex].
- Multiply the entire divisor by this quotient term:
- [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract this product from the current polynomial:
- [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
Since the remainder is zero, and all terms have been divided, the division process ends here. The quotient is [tex]\(x + 5\)[/tex], which is a polynomial.
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is the polynomial [tex]\(x + 5\)[/tex].
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