Question

The rational root of the equation 2x³ - x² - 4x + 2 = 0 is​

Answer 1

Here, we have P(x)=2x3−x2−4x+2P(x)=2x3−x2−4x+2. Let pqpq be a rational root of the polynomial. Then, p|2p|2 and q|2q|2, which implies p=±1,±2p=±1,±2 and The possible set of roots is therefore {±1,±2,±12±1,±2,±12}

Now just check for which pair (p,q)(p,q), pqpq is a root of the equation P(x)=0P(x)=0

We find that P(1/2)=0P(1/2)=0. Thus 1212 is a rational root of the given equation.

So, P(x)=2x3−x2−4x+2=2x2(x−1/2)−4(x−1/2)=(x−1/2)(2x2−4)=2(x−1/2)(x2−2)P(x)=2x3−x2−4x+2=2x2(x−1/2)−4(x−1/2)=(x−1/2)(2x2−4)=2(x−1/2)(x2−2)

Clearly, x2−2x2−2 yields no rational roots.

So, the only rational root is 1/2

Answer 2

Answer:

x^2 (2x-1) -2 (2x-1) =0. (2x-1) (x^2 -2 ) = 0. (2x-1) (x^2 -√2) (x+√2) =0. x=1/2,√2,-√2. x=1/2