Show that $1$ is not a zero of the polynomial $4 x^{4} - 3 x^{3} + 2 x^{2} - 5 x + 1$ .

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Solution

Let $p (x) = 4 x^{4} - 3 x^{3} + 2 x^{2} - 5 x + 1$
Now, $p (1) = 4 (1)^{4} - 3 (1)^{3} + 2 (1)^{2} - 5 (1) + 1 = 4 - 3 + 2 - 5 + 1 = - 1$
Clearly, $p (1) \neq 0$
Therefore, $1$ is not the zero of $4 x^{4} - 3 x^{3} + 2 x^{2} - 5 x + 1$ .
Therefore, hence showed.

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