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Answer :
To find the quotient of the polynomials [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and [tex]\(x^3 - 3\)[/tex], we need to perform polynomial long division.
### Step-by-step solution:
1. Set up the division:
- Dividend: [tex]\(x^4 + 5x^3 + 0x^2 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first term of the dividend by the first term of the divisor:
- [tex]\(x^4 \div x^3 = x\)[/tex]
- This is the first term of the quotient.
3. Multiply the entire divisor by [tex]\(x\)[/tex] and subtract from the dividend:
- Multiply: [tex]\((x^3 - 3) \times x = x^4 - 3x\)[/tex]
- Subtract: [tex]\((x^4 + 5x^3 + 0x^2 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15\)[/tex]
4. Bring down the next term and repeat:
- Now divide the first term of the new dividend by the first term of the divisor:
[tex]\[5x^3 \div x^3 = 5\][/tex]
- Multiply and subtract:
[tex]\((x^3 - 3) \times 5 = 5x^3 - 15\)[/tex]
- Subtract: [tex]\((5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0x^2 + 0x\)[/tex]
5. Since there are no more terms to bring down, the subdivision process ends.
Therefore, the quotient of dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex] is:
[tex]\[ x + 5 \][/tex]
So, the correct answer is the option labeled [tex]\(x+5\)[/tex].
### Step-by-step solution:
1. Set up the division:
- Dividend: [tex]\(x^4 + 5x^3 + 0x^2 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first term of the dividend by the first term of the divisor:
- [tex]\(x^4 \div x^3 = x\)[/tex]
- This is the first term of the quotient.
3. Multiply the entire divisor by [tex]\(x\)[/tex] and subtract from the dividend:
- Multiply: [tex]\((x^3 - 3) \times x = x^4 - 3x\)[/tex]
- Subtract: [tex]\((x^4 + 5x^3 + 0x^2 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15\)[/tex]
4. Bring down the next term and repeat:
- Now divide the first term of the new dividend by the first term of the divisor:
[tex]\[5x^3 \div x^3 = 5\][/tex]
- Multiply and subtract:
[tex]\((x^3 - 3) \times 5 = 5x^3 - 15\)[/tex]
- Subtract: [tex]\((5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0x^2 + 0x\)[/tex]
5. Since there are no more terms to bring down, the subdivision process ends.
Therefore, the quotient of dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex] is:
[tex]\[ x + 5 \][/tex]
So, the correct answer is the option labeled [tex]\(x+5\)[/tex].
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