Answer :
To find the quotient of the polynomials [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and [tex]\(x^3 - 3\)[/tex], we perform polynomial long division. Here's how you can do it step-by-step:
1. Set Up for Division:
- The divisor is [tex]\(x^3 - 3\)[/tex].
- The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/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]\(\frac{x^4}{x^3} = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], which gives you [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15\)[/tex].
4. Bring Down Terms:
- There's nothing to bring down in this step, so the remaining polynomial is simply [tex]\(5x^3 - 15\)[/tex].
5. Repeat the Process:
- Divide the new leading term by the leading term of the divisor: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the entire divisor by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
6. Check for Remainder:
- After performing these operations, the remainder is [tex]\(0\)[/tex], indicating that the division is exact.
So, the quotient of the polynomial division [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] divided by [tex]\(x^3 - 3\)[/tex] is [tex]\(x + 5\)[/tex].
1. Set Up for Division:
- The divisor is [tex]\(x^3 - 3\)[/tex].
- The dividend is [tex]\(x^4 + 5x^3 - 3x - 15\)[/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]\(\frac{x^4}{x^3} = x\)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], which gives you [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15\)[/tex].
4. Bring Down Terms:
- There's nothing to bring down in this step, so the remaining polynomial is simply [tex]\(5x^3 - 15\)[/tex].
5. Repeat the Process:
- Divide the new leading term by the leading term of the divisor: [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the entire divisor by [tex]\(5\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex].
- Subtract: [tex]\((5x^3 - 15) - (5x^3 - 15) = 0\)[/tex].
6. Check for Remainder:
- After performing these operations, the remainder is [tex]\(0\)[/tex], indicating that the division is exact.
So, the quotient of the polynomial division [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] divided by [tex]\(x^3 - 3\)[/tex] is [tex]\(x + 5\)[/tex].
