Final answer:
To divide the polynomial -2x⁴+6x³-2x²+3x-1 by x-2, we can use long division, which involves dividing the terms of the polynomial by the term of the divisor that has the highest degree. By repeating this process until there are no more terms to bring down, we can find the quotient and remainder. The final answer is -2x³+2x²-3x+2 with a remainder of 9x-1.
Step-by-step explanation:
To divide the polynomial -2x⁴+6x³-2x²+3x-1 by x-2, we can use long division.
- Write the polynomial and divisor in long division format:
- -2x⁴+6x³-2x²+3x-1 ÷ x-2
- Start by dividing the first term of the dividend (-2x⁴) by the first term of the divisor (x) to get -2x³. Write this result above the line.
- Multiply the divisor (x-2) by -2x³ and write the result underneath the dividend.
- -2x³(x-2) = -2x⁴+4x³
- Subtract this result from the original dividend to get the new dividend:
- -2x⁴+6x³-2x²+3x-1 - (-2x⁴+4x³) = 2x³-2x²+3x-1
- Repeat steps 2-4 with the new dividend until there are no more terms to bring down:
- Divide 2x³ by x to get 2x². Write this result above the line.
- Multiply the divisor (x-2) by 2x² and write the result underneath the new dividend.
- 2x²(x-2) = 2x³-4x²
- Subtract this result from the new dividend to get the new dividend:
- 2x³-2x²+3x-1 - (2x³-4x²) = 3x²+3x-1
- Divide 3x² by x to get 3x. Write this result above the line.
- Multiply the divisor (x-2) by 3x and write the result underneath the new dividend.
- 3x(x-2) = 3x²-6x
- Subtract this result from the new dividend to get the remainder:
- 3x²+3x-1 - (3x²-6x) = 9x-1
The final answer is -2x³+2x²-3x+2 with a remainder of 9x-1.