Divide 6 x to the power of 5 plus 8 x to the power of 4 plus x cubed plus x squared minus 5 x plus 1 by dividing the polynomial by 2 x cubed plus x minus 1 and find the remainder.

Step-by-step explanation:

6x⁵ + 8x⁴ + x³ + x² - 5x + 1

= (2x³ + x - 1)(Ax² + Bx + C) + (Dx + E)

= 2Ax⁵ + 2Bx⁴ + (2C + A)x³ + (B - A)x² + (C - B + D)x + (E - C)

By Comparing Coefficients, we have:

2A = 6

2B = 8

2C + A = 1

B - A = 1

C - B + D = -5

E - C = 1

Solving them, we have A = 3, B = 4, C = -1, D = 0, E = 0.

Therefore, 6x⁵ + 8x⁴ + x³ + x² - 5x + 1

= (2x³ + x - 1)(3x² + 4x - 1) + (0x + 0)

= (2x³ + x - 1)(3x² + 4x - 1).

what