Answer:
2x^2 -2x +1 remainder 6
Step-by-step explanation:
Long Division
Perhaps you want the steps of long division. Those are shown in the first attachment.
Synthetic Division
The division can also be done using synthetic division. It gets you to the same place with a lot less writing. That method is shown in the second attachment.
Graphing Calculator
A method I sometimes use when solving cubic equations is shown in the third attachment. We find the remainder value by evaluating the polynomial at x=2 (remainder theorem). Subtracting that remainder from the given polynomial gives one that is evenly divisible by (x -2). The result of that division is a quadratic that describes a parabola.
The vertex of the parabola is easily found from its graph, so the vertex form equation can be written more or less directly from the graph. Expanding that gives the quotient. (We know the leading coefficient of the quadratic is 2, because that is the leading coefficient of the given polynomial, and the leading coefficient of the divisor is 1.) For leading coefficient c and vertex (h, k), the vertex-form quadratic is ...
c(x -h)² +k
2(x -1/2)² +1/2 = 2(x² -x +1/4) +1/2 = 2x² -2x +1 . . . quotient
we already know the remainder is 6
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Additional comment
At each step of the long division, you multiply the quotient term by the divisor and subtract that product. The net result is that the highest-degree dividend term is cancelled.
Synthetic division makes this a little easier by doing two things. It ignores the arithmetic with the most-significant terms of quotient and divisor, because the result of that arithmetic is always zero. And, it uses the opposite of the binomial divisor constant for doing the multiplication, so the result can be added instead of subtracted. (Addition is generally easier to do without making mistakes.) You notice the number at upper left in the synthetic division is 2, not -2, for division by (x-2).
