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 Question from archit, a student: If P(x)=x^4+ax^3+bx^2-8x+1 is a perfect square then (a+b)=?

Hi Archit,

If $P(x)$ is a square it must be of the form $\left(c_1 x^2 + c_2 x + c_3\right)^2.$ Thus you have

$x^4 + a x^3 + b x^2 - 8 x + 1 = \left(c_1 x^2 + c_2 x + c_3\right)^2.$

Expand the right side and compare the coefficients on the right and left sides. What are the possible values of $c_1?$ What are the possible values of $c_3?$ Given these values what can $c_2$ be? Now that you know the possible values of the coefficients on the right side what can you say about $a \mbox{and} b?$

Penny

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