Rational roots of the equation $2x^4+x^3-11x^2+x+2=0$ are
- $\dfrac {1}{2}, 2$
- $\dfrac {1}{2}, 2, \dfrac {1}{4}, -2$
- $\dfrac {1}{2}, 2, 3, 4$
- $\dfrac {1}{2}, 2, \dfrac {3}{4}, -2$
Using the Rational Root Theorem, possible roots are factors of 2 divided by factors of 2, i.e., +/- 1, +/- 2, +/- 1/2. Testing x = 1/2: 2(1/16) + 1/8 - 11/4 + 1/2 + 2 = 1/8 + 1/8 - 22/8 + 4/8 + 16/8 = 0. Testing x = 2: 2(16) + 8 - 11(4) + 2 + 2 = 32 + 8 - 44 + 4 = 0. Both are roots.
By the rational root theorem, any rational root p over q of the polynomial must have p dividing the constant 2 and q dividing the leading coefficient 2. Testing x equals 2 gives 32 plus 8 minus 44 plus 2 plus 2, which equals 0, and testing x equals one half gives one eighth plus one eighth minus eleven fourths plus one half plus 2, which also equals 0. Performing polynomial division by (x minus 2) and (2x minus 1) leaves the quadratic x squared plus 2x minus 1, whose roots are irrational. Thus, the only rational roots are one half and 2.