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Exercise 2.3

1.  Find the remainder when x³ + 3x³ + 3x + 1 is divided by
(i) x + 1
Sol. Given : let p(x) = x³ + 3x² + 3x + 1.

Ans.  –  When p(x) is divided by x+1, teh remainder is p(-1).
∴ p(-1) = (-1)3 +3(-1)2 + 3(-1) + 1
= -1 + 3 -3 + 1 = 0

(ii) $x - {1 \over 2}$

Ans.  –  When p(x) is divided by $\left( {x - {1 \over 2}} \right)$, the remainder is $p\left( {{1 \over 2}} \right)$.
$p\left( {{1 \over 2}} \right) = {\left( {{1 \over 2}} \right)^3} + 3{\left( {{1 \over 2}} \right)^2} + \left( {{1 \over 2}} \right) + 1$
$= {1 \over 8} + {3 \over 4} + {3 \over 2} + 1$
$= {{ - 125 + 150 - 60 + 8} \over 8} = {{ - 27} \over 8}$

(iii) x

Ans.  –  when p(x) is divided by x, the remainder is p(0).
∴ p(0) = 0 + 0 + 0 + 1  = 1.

(iv) x + π

Ans.  –  When p(x) is divided by 5 + 2x, the remainder is   p(-π).
∴ p(-π) =  (-π)³ +3(-π)² + 3 (-π) + 1
= -π³ + 3π² – 3π + 1

(v) 5 + 2x.

Ans.  –  When p(x) is divided by 5 + 2x, the remainder is $p\left( { - {5 \over 2}} \right)$.
$p\left( { - {5 \over 2}} \right) = {\left( {{5 \over 2}} \right)^3} + 3{\left( { - {5 \over 2}} \right)^2} + 3\left( { - {5 \over 2}} \right) + 1$
$= - {{125} \over 8} + {{75} \over 4} + {{15} \over 2} + 1$
$= {{ - 125 + 150 - 60 + 8} \over 8} = {{ - 27} \over 8}$

2. Find the remainder when x³-ax²+6x-a id divided by x-a.

Sol.   When p(x) = x³ – ax² + 6x – a is divided by (x-a), the remainder is p(a).
∴ p(a) = a³ – a³ + 6a – a = 5a.

3. Check whether 7 + 3x is a factor of 3x³ + 7x.

Sol.  If 7 + 3x is a factor of 3x³ + 7x, then $p\left( { - {7 \over 3}} \right) = 0$
$p\left( { - {7 \over 3}} \right) = 3 \times {\left( {{{ - 7} \over 3}} \right)^3} + 7 \times {{( - 7)} \over 3}$
$= 3 \times {{ - 343} \over {27}} - {{49} \over 3} = {{ - 343} \over 9} - {{49} \over 3}$
$= {{ - 343 - 147} \over 9} = {{ - 490} \over 9} \ne 0.$