The polynomial x^4 - 27x^2 + 121 can be factored in a unique way into a product of two quadratic polynomials with integer coefficients and leading coefficient 1. What is the sum of these two polynomials?
+128475 x^4 - 27x^2 + 121
Let us guess that the factorization may be
(x^2 + ax + 11) (x^2 - ax + 11) expand
x^4 - ax^3 + 11x^2 + ax^3 - a^2x^2 + 11ax + 11x^2 - 11ax + 121 simplify
x^4 + 22x^2 - a^2x^2 + 121
Note that, equating coefficients, (22 - a^2) = -27 rearrange
22 + 27 = a^2
49 = a^2
Take both roots
7 = a and -7 = a
So we have
( x^2 + 7x + 11 ) (x^2 - 7x + 11)
The sum is
2x^2 + 22
