Answer :
To solve this problem, we need to find the quotient when the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] is divided by [tex]\(x^3 - 3\)[/tex]. We'll do this using polynomial long division. Let's go through each step of the division process.
1. Setup the Division:
- Divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. First Division Step:
- Divide the leading term of the numerator [tex]\(x^4\)[/tex] by the leading term of the denominator [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
- Multiply [tex]\(x\)[/tex] by the entire denominator: [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original polynomial: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15\)[/tex].
3. Second Division Step:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]. This gives [tex]\(+5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the entire denominator: [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the result [tex]\(5x^3 + 0x - 15\)[/tex]: [tex]\((5x^3 + 0x - 15) - (5x^3 - 15) = 0x + 0\)[/tex].
4. Conclusion:
- The remainder is zero, which means that division results in the quotient [tex]\(x + 5\)[/tex].
Hence, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Correct option: [tex]\(x + 5\)[/tex].
1. Setup the Division:
- Divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. First Division Step:
- Divide the leading term of the numerator [tex]\(x^4\)[/tex] by the leading term of the denominator [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
- Multiply [tex]\(x\)[/tex] by the entire denominator: [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original polynomial: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15\)[/tex].
3. Second Division Step:
- Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]. This gives [tex]\(+5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the entire denominator: [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from the result [tex]\(5x^3 + 0x - 15\)[/tex]: [tex]\((5x^3 + 0x - 15) - (5x^3 - 15) = 0x + 0\)[/tex].
4. Conclusion:
- The remainder is zero, which means that division results in the quotient [tex]\(x + 5\)[/tex].
Hence, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
Correct option: [tex]\(x + 5\)[/tex].
