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Answer :
To solve the problem of finding the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], we will perform polynomial division.
### Step-by-Step Solution
1. Identify the Divisor and Dividend:
- The polynomial we want to divide is the dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor is the polynomial we're dividing by: [tex]\(x^3 - 3\)[/tex].
2. Set Up the Division:
- We are dividing a degree 4 polynomial by a degree 3 polynomial, which means the result will be a degree [tex]\(4 - 3 = 1\)[/tex] polynomial.
3. Perform the Division:
- First Term of Quotient:
- 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].
- Multiply [tex]\(x\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
- Subtract this from the original polynomial: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
- Next Term of Quotient:
- Divide the leading term of the remaining polynomial [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract this from the remainder: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
The division terminates here with no remainder.
4. Check the Results:
- The quotient of the division is [tex]\(x + 5\)[/tex].
- Since the remainder is 0, the division is exact.
Therefore, the quotient when [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] is divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
### Step-by-Step Solution
1. Identify the Divisor and Dividend:
- The polynomial we want to divide is the dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
- The divisor is the polynomial we're dividing by: [tex]\(x^3 - 3\)[/tex].
2. Set Up the Division:
- We are dividing a degree 4 polynomial by a degree 3 polynomial, which means the result will be a degree [tex]\(4 - 3 = 1\)[/tex] polynomial.
3. Perform the Division:
- First Term of Quotient:
- 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].
- Multiply [tex]\(x\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
- Subtract this from the original polynomial: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
- Next Term of Quotient:
- Divide the leading term of the remaining polynomial [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the divisor [tex]\(x^3 - 3\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract this from the remainder: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
The division terminates here with no remainder.
4. Check the Results:
- The quotient of the division is [tex]\(x + 5\)[/tex].
- Since the remainder is 0, the division is exact.
Therefore, the quotient when [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] is divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
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