2011/096) The equation P(x) = x^4 – 16x^3 + 94x^2 +px + q = 0 has two double roots. Solve

Because it has 2 double roots so it is square of a quadratic polynomial

Let it be (x^2+ax+b)^2

= x^4 + 2ax^3 + x^2(2b+a^2) + 2abx+ b^2

Comparing coefficients

2x = -16 or a = - 8

2b+a^2 = 94 or b = 15

p = 2ab = - 240

q = b^2 = 225

so equation

= x^4-16x^2 + 94x^2 – 240x + 225

It is (x^2 -8 x + 15) ^2 = (x-3)^2(x-5)^2 and roots are x = 3,3,5,5