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Let a and b real numbers such that x^4+2x^3-4x^2+ax+b = (Q(x))^2 for some polynomial Q(x). What is the value of a + b?

 Jun 14, 2021
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since the final polynomial is a fourth-degree polynomial, Q(x) must be quadratic, and since it is monic, Q(x) is also monic. Therefore, Q(x) can be expressed as x^2+cx+d, where c and d are real numbers. If you were to square it, you would get x^4 + 2 c x^3 + c^2 x^2 + 2 d x^2 + 2 c d x + d^2. Therefore, we can solve it using equations by matching the coefficients:

\(2c=1\\c=1\\2d+c^2=2d+1=-4\\d=-2.5\)

Now just match the coefficients once again to solve for a and b, and then add them up. Can you finish it?

 Jun 15, 2021

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