Answer :
To solve the polynomial division problem [tex]\((6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16) \div (x^2 - x + 2)\)[/tex], we'll perform polynomial long division. Follow these steps:
1. Set up the division: Place the dividend [tex]\(6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16\)[/tex] under the division symbol and the divisor [tex]\(x^2 - x + 2\)[/tex] outside.
2. Divide the leading terms: Divide the leading term of the dividend [tex]\(6x^5\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(6x^3\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(6x^3\)[/tex] by the entire divisor [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(6x^5 - 6x^4 + 12x^3\)[/tex].
- Subtract this result from the current dividend to get the new polynomial: [tex]\((6x^5 + 4x^4 + 11x^3) - (6x^5 - 6x^4 + 12x^3) = 10x^4 - x^3\)[/tex].
4. Repeat the process:
- Divide the new leading term [tex]\(10x^4\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], giving [tex]\(10x^2\)[/tex].
- Multiply [tex]\(10x^2\)[/tex] by [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(10x^4 - 10x^3 + 20x^2\)[/tex].
- Subtract: [tex]\((10x^4 - x^3) - (10x^4 - 10x^3 + 20x^2) = 9x^3 + 3x^2\)[/tex].
5. Continue dividing:
- Divide [tex]\(9x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(9x\)[/tex].
- Multiply [tex]\(9x\)[/tex] by [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(9x^3 - 9x^2 + 18x\)[/tex].
- Subtract: [tex]\((9x^3 + 3x^2) - (9x^3 - 9x^2 + 18x) = 12x^2 - 6x\)[/tex].
6. Final division:
- Divide [tex]\(12x^2\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(12\)[/tex].
- Multiply [tex]\(12\)[/tex] by [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(12x^2 - 12x + 24\)[/tex].
- Subtract: [tex]\((12x^2 - 6x) - (12x^2 - 12x + 24) = 6x - 8\)[/tex].
7. Conclusion:
- The division is complete when the degree of the remainder ([tex]\(6x - 8\)[/tex]) is less than the degree of the divisor ([tex]\(x^2 - x + 2\)[/tex]).
- The quotient is [tex]\(6x^3 + 10x^2 + 9x + 12\)[/tex] and the remainder is [tex]\(6x - 8\)[/tex].
So, the solution to the division problem is:
[tex]\[ \text{Quotient: } 6x^3 + 10x^2 + 9x + 12 \][/tex]
[tex]\[ \text{Remainder: } 6x - 8 \][/tex]
1. Set up the division: Place the dividend [tex]\(6x^5 + 4x^4 + 11x^3 + 23x^2 + 12x + 16\)[/tex] under the division symbol and the divisor [tex]\(x^2 - x + 2\)[/tex] outside.
2. Divide the leading terms: Divide the leading term of the dividend [tex]\(6x^5\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], which gives [tex]\(6x^3\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(6x^3\)[/tex] by the entire divisor [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(6x^5 - 6x^4 + 12x^3\)[/tex].
- Subtract this result from the current dividend to get the new polynomial: [tex]\((6x^5 + 4x^4 + 11x^3) - (6x^5 - 6x^4 + 12x^3) = 10x^4 - x^3\)[/tex].
4. Repeat the process:
- Divide the new leading term [tex]\(10x^4\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex], giving [tex]\(10x^2\)[/tex].
- Multiply [tex]\(10x^2\)[/tex] by [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(10x^4 - 10x^3 + 20x^2\)[/tex].
- Subtract: [tex]\((10x^4 - x^3) - (10x^4 - 10x^3 + 20x^2) = 9x^3 + 3x^2\)[/tex].
5. Continue dividing:
- Divide [tex]\(9x^3\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(9x\)[/tex].
- Multiply [tex]\(9x\)[/tex] by [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(9x^3 - 9x^2 + 18x\)[/tex].
- Subtract: [tex]\((9x^3 + 3x^2) - (9x^3 - 9x^2 + 18x) = 12x^2 - 6x\)[/tex].
6. Final division:
- Divide [tex]\(12x^2\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(12\)[/tex].
- Multiply [tex]\(12\)[/tex] by [tex]\(x^2 - x + 2\)[/tex] to get [tex]\(12x^2 - 12x + 24\)[/tex].
- Subtract: [tex]\((12x^2 - 6x) - (12x^2 - 12x + 24) = 6x - 8\)[/tex].
7. Conclusion:
- The division is complete when the degree of the remainder ([tex]\(6x - 8\)[/tex]) is less than the degree of the divisor ([tex]\(x^2 - x + 2\)[/tex]).
- The quotient is [tex]\(6x^3 + 10x^2 + 9x + 12\)[/tex] and the remainder is [tex]\(6x - 8\)[/tex].
So, the solution to the division problem is:
[tex]\[ \text{Quotient: } 6x^3 + 10x^2 + 9x + 12 \][/tex]
[tex]\[ \text{Remainder: } 6x - 8 \][/tex]
