Which number added to the polynomial $3 x^{2} - 4 x - 1$ gives a polynomial with $(x - 1)$ as a factor?

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Solution

$p (x) = 3 x^{2} - 4 x - 1$
$p (1) = 3 (1)^{2} - 4 (1)^{2} - 1 = 3 - 4 - 1 = - 2$
Thus, when the polynomial $3 x^{2} - 4 x - 1$ is divided by $(x - 1)$ , the remainder is -2.
And, hence, 2 should be added to the given polynomial so that $(x - 1)$ is a factor of it.

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