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Question

If the expression ax4+bx3x2+2x+3 has remainder 4x+3 when divided by x2+x2, then which of the following is/are true?

A
a=2
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B
b=2
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C
a=1
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D
b=1
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Solution

The correct option is C a=1
Let f(x)=ax4+bx3x2+2x+3
Using remainder theorem,
f(x)=(x2+x2)Q(x)+4x+3
f(x)=(x+2)(x1)Q(x)+4x+3

f(2)=4(2)+3=5
a(2)4+b(2)3(2)2+2(2)+3=8+3
16a8b44+3=5
16a8b5=5
16a8b=0
2a=b .....(i)

And f(1)=4(1)+3
a(1)4+b(1)3(1)2+2(1)+3=4+3
a+b1+2+3=7
a+b=3 .....(ii)

On solving (i) and (ii),
2a=b and a+b=3,
we get a=1,b=2

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