Answer :
To find the quotient of the polynomials [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and [tex]\(x^3 - 3\)[/tex], we are going to perform polynomial long division. Here's how it works:
1. Identify the dividend and divisor:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Perform the division:
- Step 1: Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex] to get the first term of the quotient, which is [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this term [tex]\(x\)[/tex] and subtract the result from the dividend.
- Multiply: [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex]
- Subtract: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x - 15\)[/tex]
- Step 3: Repeat the process with the new dividend [tex]\(5x^3 - 0x - 15\)[/tex].
- Divide the first term [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply: [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex]
- Subtract: [tex]\((5x^3 - 0x - 15) - (5x^3 - 15) = 0\)[/tex]
3. Conclusion:
- The quotient from the division is [tex]\(x + 5\)[/tex].
- Since the remainder is zero, our division is complete.
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Identify the dividend and divisor:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Perform the division:
- Step 1: Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex] to get the first term of the quotient, which is [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this term [tex]\(x\)[/tex] and subtract the result from the dividend.
- Multiply: [tex]\(x \times (x^3 - 3) = x^4 - 3x\)[/tex]
- Subtract: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 0x - 15\)[/tex]
- Step 3: Repeat the process with the new dividend [tex]\(5x^3 - 0x - 15\)[/tex].
- Divide the first term [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply: [tex]\(5 \times (x^3 - 3) = 5x^3 - 15\)[/tex]
- Subtract: [tex]\((5x^3 - 0x - 15) - (5x^3 - 15) = 0\)[/tex]
3. Conclusion:
- The quotient from the division is [tex]\(x + 5\)[/tex].
- Since the remainder is zero, our division is complete.
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
