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Answer :
To solve the problem of finding the quotient when dividing the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we'll perform polynomial long division. Here's a step-by-step explanation:
1. Setup the Polynomial Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. 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]).
- [tex]\(x^4 \div x^3 = x\)[/tex]
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(x\)[/tex] and subtract from the dividend.
- [tex]\( (x^3 - 3) \cdot x = x^4 - 3x \)[/tex]
- Subtract from the dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15\)[/tex]
4. Repeat:
- Now repeat the process with the new dividend [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by the leading term [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex]
5. Multiply and Subtract again:
- Multiply the entire divisor by 5 and subtract:
- [tex]\( (x^3 - 3) \cdot 5 = 5x^3 - 15 \)[/tex]
- Subtract: [tex]\((5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0\)[/tex]
6. Result:
- Since we have no remainder and have accounted for all terms in the original polynomial, the quotient is simply:
- Quotient: [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].
The correct answer is:
[tex]\[ x - 5 \][/tex]
1. Setup the Polynomial Division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. 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]).
- [tex]\(x^4 \div x^3 = x\)[/tex]
3. Multiply and Subtract:
- Multiply the entire divisor by [tex]\(x\)[/tex] and subtract from the dividend.
- [tex]\( (x^3 - 3) \cdot x = x^4 - 3x \)[/tex]
- Subtract from the dividend: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15\)[/tex]
4. Repeat:
- Now repeat the process with the new dividend [tex]\(5x^3 + 0x^2 + 0x - 15\)[/tex].
- Divide the leading term [tex]\(5x^3\)[/tex] by the leading term [tex]\(x^3\)[/tex].
- [tex]\(5x^3 \div x^3 = 5\)[/tex]
5. Multiply and Subtract again:
- Multiply the entire divisor by 5 and subtract:
- [tex]\( (x^3 - 3) \cdot 5 = 5x^3 - 15 \)[/tex]
- Subtract: [tex]\((5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0\)[/tex]
6. Result:
- Since we have no remainder and have accounted for all terms in the original polynomial, the quotient is simply:
- Quotient: [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].
The correct answer is:
[tex]\[ x - 5 \][/tex]
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