+78 Let a and b real numbers such that x4+2x3-x2+ax+b = (Q(x))2 for some polynomial Q(x). What is the value of a + b?
+6251 \(p(x)=x^4+2x^3-x^2+ax+b = (q(x))^2,~\text{for some polynomial }q(x)\)
\(\text{we know q(x) will be of order 2 so write it as }\\ q(x)=q_2 x^2 + q_1 x + q_0 \\ (q(x))^2 = q_2^2 x^4+2 q_1 q_2 x^3+\left(q_1^2+2 q_0 q_2\right) x^2+2 q_0 q_1 x+q_0^2 \)
\(\text{and we have equations }\\ q_2^2 = 1\\ 2q_1q_2 = 2\\ (q_1^2+2q_0q_2)=-1\\ 2q_0q_1=a\\ q_0^2=b\)
\(\text{clearly }q_2 = \pm 1\\ \text{suppose }q_2=1 \\ 2q_1 (1)=2,~q_1=1\\ (1+2q_0(1))=-1 \\ q_0=-1 \\ b=q_0^2 = 1 \\ a=2q_0 = 2\)
\(\text{now suppose }q_2=-1 \\ 2q_1(-1) = 2,~q_1=-1 \\ (1+2q_0(-1))=-1,~q_0=1 \\ b=q_0^2 = 1 \\ a = 2(-1)(-1) = 2\)
\(\text{so in both cases }a=2,~b=1 \\ \text{and }a+b = 3\)
I kind of suspect there is a simpler way to solve this.
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