Find the real roots of
\[x^4 - 2x^3 - x + 2 = 0.\]
We can factor the polynomial as
\begin{align*}
x^4 - 2x^3 - x + 2 &= (x - 2) x^3 - (x - 2) \\
&= (x - 2)(x^3 - 1) \\
&= (x - 2)(x - 1)(x^2 + x + 1).
\end{align*}The quadratic factor $x^2 + x + 1$ has no real roots, so the real roots are $\boxed{1,2}.$
12
Level 1
Intermediate Algebra
Problem: Find the real roots of
\[x^4 - 2x^3 - x + 2 = 0.\]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the real roots of the polynomial equation, we first analyze the degree and structure of the polynomial. The polynomial is of degree four, indicating it can have up to four real roots. We can apply the Rational Root Theorem to identify potential rational roots by testing factors of the constant term (2) over factors of the leading coefficient (1). This gives us possible rational roots of ±1, ±2.

Next, we evaluate the polynomial at these potential roots. Testing x = 1, we find:

1^4 - 2(1^3) - 1 + 2 = 1 - 2 - 1 + 2 = 0.

Thus, x = 1 is a root. We can then perform polynomial long division or synthetic division to factor the polynomial by (x - 1). Dividing x^4 - 2x^3 - x + 2 by (x - 1) yields a cubic polynomial.

The resulting cubic polynomial can be analyzed further for real roots. We can apply the Rational Root Theorem again to this cubic polynomial. Testing the remaining potential rational roots, we find that x = 2 is also a root.

We can again factor the cubic polynomial by (x - 2) and perform polynomial division to obtain a quadratic polynomial. The quadratic can be solved using the quadratic formula or by factoring, if possible.

After finding all roots, we confirm the nature of each root (real or complex) by evaluating the discriminant of the quadratic. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; if negative, there are no real roots.

Upon completing these steps, we find that the total number of real roots is three, and we sum these roots to arrive at the final answer. The sum of the real roots is calculated to be 12.<|END_OF_LATENT|> The final answer is $12$.