First slide

Theory of equations

Question

$When an unknown polynomial is divided by (x - 1) and (x - 2) we obtain the remainder 2 and 1, respectively.$
$Then the remainder resulting from the division of this polynomial by (x - 1) (x - 2) is$

Moderate

A

$3 - x$
B

$2 x - 3$
C

$2 + 3 x$
D

$x - 2$

Solution

$Let P (x) be the unknown polynomial and let q (x) and r (x) = a x + b be the quotient and the remainder, respectively,$
$resulting from the division of that polynomial by (x - 1) (x - 2) .$
$\begin{aligned} \Rightarrow P (x) = (x - 1) (x - 2) q (x) + a x + b \\ from the problem P (1) = 2 \Rightarrow a + b = 2 \end{aligned}$
$\begin{array}{l} Also P (2) = 1 \Rightarrow 2 a + b = 1 \Rightarrow a = - 1, b = 3 \\ ∴ Required remainder = (3 - x) \end{array}$

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