Question

f(x) = x4  2x3 + 3x2  ax + b leaves remainder 5 and 19 on division by (x  1) and
(x + 1) respectively. Find the remainder when f(x) is divided by (x  2).
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Answer 1

Since u have not written symbols I'm considering the question is--F(x) = x4 + 2x3 + 3x2 + ax + b leaves remainder 5 and 19 on division by (x - 1) and (x + 1) respectively. Find the remainder when f(x) is divided by (x + 2). So ,when divided by (x-1) ,x=1. Substituting this value in F(x) should give 5 as per the remainder theorem.. x⁴+2x³+3x²+ax+b=5 --->1+2+3+a+b=5 --->a+b=-1....(eq1) When ,x=(-1) x⁴+2x³+3x²+ax+b=19 --->1-2+3-a+b=19 -->-a+b=19-1+2-3 -->. -a+b=17...eq(2) Adding eq 1 eq 2 ,2b=16....b=8 Therefore ,a=-9 Now the equation becomes x⁴+2x³+3x²-9x+8 So when divided by (x+2),x=(-2) (-2)⁴+2(-2)³+3(-2)²-9×(-2)+8 =16-16+12+18+8 =38 The remainder will be 38