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Question

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) p(x)=x33x2+5x3, g(x)=x22
(ii) p(x)=x43x2+4x+5, g(x)=x2+1x
(iii) p(x)=x45x+6, g(x)=2x2

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Solution

(i) x3
x22)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x33x2+5x3
x3 2x
________________
3x2+7x3
3x2 6
________________
7x9
Quotient=x3
Remainder=7x9

(ii) x2+x3
x2x+1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x43x2+4x+5+0x3
x4+x2 x3
______________________
x3+4x2+4x+5
x3x2+x
______________________
3x2+3x+5
3x2+3x3
______________________
8
Quotient=x2+x3
Remainder=8

(iii) x22
2x2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯x4+0x3+0x25x+6
x4 2x2
____________________
0x3+2x25x+6
+2x2 4
_____________________
5x+10
Quotient=x22
Remainder=5x+10.


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