Answer :
To find the quotient and remainder of the polynomial division [tex]\((2x^4 + 5x^3 - 2x - 8) \div (x+3)\)[/tex], we need to perform polynomial long division. Here's how it's done step-by-step:
1. Divide the leading term of the dividend by the leading term of the divisor:
- Leading term of dividend: [tex]\(2x^4\)[/tex]
- Leading term of divisor: [tex]\(x\)[/tex]
- Divide: [tex]\(2x^4 \div x = 2x^3\)[/tex]
2. Multiply the entire divisor by this result and subtract:
- Multiply: [tex]\((x + 3) \times 2x^3 = 2x^4 + 6x^3\)[/tex]
- Subtract: [tex]\((2x^4 + 5x^3 - 2x - 8) - (2x^4 + 6x^3) = -x^3 - 2x - 8\)[/tex]
3. Repeat the process with the new polynomial:
- Leading term: [tex]\(-x^3\)[/tex]
- Divide: [tex]\(-x^3 \div x = -x^2\)[/tex]
4. Multiply and subtract again:
- Multiply: [tex]\((x + 3) \times -x^2 = -x^3 - 3x^2\)[/tex]
- Subtract: [tex]\((-x^3 - 2x - 8) - (-x^3 - 3x^2) = 3x^2 - 2x - 8\)[/tex]
5. Continue the process:
- Leading term: [tex]\(3x^2\)[/tex]
- Divide: [tex]\(3x^2 \div x = 3x\)[/tex]
6. Multiply and subtract:
- Multiply: [tex]\((x + 3) \times 3x = 3x^2 + 9x\)[/tex]
- Subtract: [tex]\((3x^2 - 2x - 8) - (3x^2 + 9x) = -11x - 8\)[/tex]
7. Final step:
- Leading term: [tex]\(-11x\)[/tex]
- Divide: [tex]\(-11x \div x = -11\)[/tex]
8. Multiply and subtract:
- Multiply: [tex]\((x + 3) \times -11 = -11x - 33\)[/tex]
- Subtract: [tex]\((-11x - 8) - (-11x - 33) = 25\)[/tex]
Finally, the quotient is [tex]\(2x^3 - x^2 + 3x - 11\)[/tex] and the remainder is 25.
Thus, the correct option is c. [tex]\(2x^3 - x^2 + 3x - 11\)[/tex]; 25.
1. Divide the leading term of the dividend by the leading term of the divisor:
- Leading term of dividend: [tex]\(2x^4\)[/tex]
- Leading term of divisor: [tex]\(x\)[/tex]
- Divide: [tex]\(2x^4 \div x = 2x^3\)[/tex]
2. Multiply the entire divisor by this result and subtract:
- Multiply: [tex]\((x + 3) \times 2x^3 = 2x^4 + 6x^3\)[/tex]
- Subtract: [tex]\((2x^4 + 5x^3 - 2x - 8) - (2x^4 + 6x^3) = -x^3 - 2x - 8\)[/tex]
3. Repeat the process with the new polynomial:
- Leading term: [tex]\(-x^3\)[/tex]
- Divide: [tex]\(-x^3 \div x = -x^2\)[/tex]
4. Multiply and subtract again:
- Multiply: [tex]\((x + 3) \times -x^2 = -x^3 - 3x^2\)[/tex]
- Subtract: [tex]\((-x^3 - 2x - 8) - (-x^3 - 3x^2) = 3x^2 - 2x - 8\)[/tex]
5. Continue the process:
- Leading term: [tex]\(3x^2\)[/tex]
- Divide: [tex]\(3x^2 \div x = 3x\)[/tex]
6. Multiply and subtract:
- Multiply: [tex]\((x + 3) \times 3x = 3x^2 + 9x\)[/tex]
- Subtract: [tex]\((3x^2 - 2x - 8) - (3x^2 + 9x) = -11x - 8\)[/tex]
7. Final step:
- Leading term: [tex]\(-11x\)[/tex]
- Divide: [tex]\(-11x \div x = -11\)[/tex]
8. Multiply and subtract:
- Multiply: [tex]\((x + 3) \times -11 = -11x - 33\)[/tex]
- Subtract: [tex]\((-11x - 8) - (-11x - 33) = 25\)[/tex]
Finally, the quotient is [tex]\(2x^3 - x^2 + 3x - 11\)[/tex] and the remainder is 25.
Thus, the correct option is c. [tex]\(2x^3 - x^2 + 3x - 11\)[/tex]; 25.
