The polynomial $p (x) = 4 x^{3} + 7 x^{2} - 5 x + t$ , when divided by $(x + 1)$ gives 3 as the remainder. Find the value of $t$ .

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Solution

Using the remainder theorem, the remainder when p(x) is divided by (x+1) is p(-1).
The value of the remainder is given as 3.
The remainder when P(x) is divided by (x + 1) is 3.
$∴ p (- 1) = 3$
$\Rightarrow 4 \times (- 1)^{3} + 7 \times (- 1)^{2} - 5 (- 1) + t = 3$
$4 \times (- 1) + 7 \times 1 + 5 + t = 3$
$- 4 + 7 + 5 + t = 3$
$t = - 5$

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Find the value of ‘t’ for which the polynomial P(x) = $x^{3} + 3 x^{2} + 2 x + t$ gives 3 as remainder when divided by( x – 1).

Statement1: The polynomial $P (x) = 4 x^{3} - 3 x^{2} + 5 x - 6$ when divided by $x - 1$ gives zero as the remainder.

Statement2: $(x - 1)$ is a factor of the polynomial $P (x) = 4 x^{3} - 3 x^{2} + 5 x - 6$ .

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