Question

Let abpq Ïµ Q and suppose that f (x) = x2 + ax + b = 0 and g (x) = x3px + q = 0 have a common irrational root, then

Solution

Correct option is

f (x) divides g (x)

Let α Ïµ R – Q be a common root of f (x) = 0 and g (x) = 0. Then α2 = –aα – b. substituting this in a3 + pα + q = 0, we get

                      (a2 – b + p) α + ab + q = 0

As α is irrational and abpq Ïµ Qp = b – a2q = – ab. This gives, g(x) = (x – af (x).

SIMILAR QUESTIONS

Q1

If three distinct real number ab and c satisfy

                                         

Where p Ïµ R, then value of b c + ca + a b is

Q2

The number of integral roots of the equation 

is

Q3

The product of roots of 

Q4

Let S denote the set of all values of the parameter a for which

                        

has no solution, then S equals

Q5

The number of roots of the equation 

Q6

If all the roots of x3 + px + q = 0 pq Ïµ R q ≠ 0 are real, then

Q7

Let S denote the set of all real value of q for which the roots of the equation

                          x2 – 2ax + a2 – 1 = 0          ...(1)

lie between 5 and 10, then S equals

Q8

Let S denote the set of all values of a for which the roots of the equation (1 + a)x2 – 3ax + 4a = 0 exceed 1, then S equals

Q9

Let S denote the set of all values of S for which the equation

         2x2 – 2(2a + 1) x + a(a + 1) = 0

has one root less than a and root greater than a, then S equals

Q10

The number of irrational solutions of the equation