The number of positive integral roots of $x^{4} + x^{3} - 4 x^{2} + x + 1 = 0$ is/are

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Solution

The equation is $x^{4} + x^{3} - 4 x^{2} + x + 1 = 0$
$\Rightarrow (x^{4} - 2 x^{2} + 1) + (x^{3} - 2 x^{2} + x) = 0$
$\Rightarrow (x^{2} - 1)^{2} + x (x^{2} - 2 x + 1) = 0$
$\Rightarrow (x - 1)^{2} {(x + 1)^{2} + x} = 0$
$\Rightarrow (x - 1)^{2} = 0$ or $x^{2} + 3 x + 1 = 0$
Roots are $1, 1, \frac{- 3 + \sqrt{5}}{2}, \frac{- 3 - \sqrt{5}}{2}$
Positive roots are $1, 1$ .
Number of positive roots are $2$ .

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