Cubic + Quadratic = Choked up with less

Algebra Level pending

Suppose a cubical polynomial x 3 + p x 2 + q x + 72 x^{3} + p x^{2} + qx + 72 such that it is divisible by x 2 + a x + b x^{2} + ax+ b and x 2 + b x + a x^{2} + bx+ a . Here a is not equal to b. And a , b , q , p a,b,q,p are constants . Find sum of squares of the roots of cubic polynomial + a + b + a b + 72 a + b + ab + 72


The answer is 145.

Aakash Khandelwal by Aakash Khandelwal , 21, India

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1 solution

Since 1 is common root of both quadratic equations 1 is also a root of cubic. Hence p+q = -73. Also other roots are b and a respectively of quadratic equations respectively. Therefore by Vietas a+b = -1 . Now p= -( 1 + a+ b) = 0.
Therefore q= -73 . Now sum if squares of roots if cubic = p^2 -q = 146 . Since sum of a and b is minus one and their product is -72. ANSWER=145

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