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Question 2
The polynomial p(x)=x42x3+3x2ax+3a7 when divided by x + 1 leaves the remainder 19. Find the value of a. Also , find the remainder when p(x) is divided by x + 2.

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Solution

Given,
p(x)=x42x3+3x2ax+3a7
When we divide p(x) by x+1, then we get the remainder as p(- 1).
P(1)=(1)42(1)3+3(1)2a(1)+3a7
1+2+3+a+3a7=4a1
According to the question, p(-1) = 19
4a1=19
4a=20
a=5
Required polynomial =x42x3+3x25x+3(5)7
=x42x3+3x25x+157
=x42x3+3x25x+8
When we divide p(x) by x + 2, we get the remainder as p(-2).
Now, p(2)=(2)42(2)3+3(2)25(2)+8
=16+16+12+10+8=62

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