Question

10. The rational root of the equation 2x^3- x^2-4x -+2 = 0 is
1
c)2
2
a)
b) -​

Answer 1

Answer:

12

Step-by-step explanation:

If the polynomial P(x)=anxn+an−1xn−1+...+a2x2+a1x+a0 possesses any rational root pq , where p,q∈Z with g.c.d(p,q)=1 , then p divides a0 and q divides an . In mathematical notation, p|a0 and q|an

Here, we have P(x)=2x3−x2−4x+2 . Let pq be a rational root of the polynomial. Then, p|2 and q|2 , which implies p=±1,±2 and q=±1,±2

The possible set of roots is therefore { ±1,±2,±12 }

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

We find that P(1/2)=0 . Thus 12 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)

Clearly, x2−2 yields no rational roots.

So, the only rational root is 12

Answer 2

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