Answer:
remainder: 3/4
Step-by-step explanation:
You want to divide the polynomial (2x^4-9x^3-x^2+2) by (2x-1).
Polynomial long division
The attachment shows the division worked out.
Your work has a sign error in the second product: (-4x²)(2x-1) should be (-8x³+4x²).
Factor theorem
The factor theorem tells you that if division by (2x-1) gives a remainder of 0, then (2x-1) is a factor of the dividend polynomial. Alas, here, the remainder is not zero, so 2x-1 is not a factor.
Remainder theorem
The remainder theorem tells you that the remainder of p(x) divided by (x-a) is p(a). We can evaluate the polynomial for x=1/2, the value that makes the divisor zero:
f(1/2) = ((2(1/2) -9)(1/2) -1)(1/2)^2+2 = ((-8)(1/2) -1)(1/4) +2 =(-5)(1/4) +2 = 3/4
The remainder from division by (2x -1) is 3/4.
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Additional comment
We did the polynomial evaluation by writing it in Horner form. This minimizes the number of arithmetic operations required, and generally makes them simpler.
