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Question

# If x3âˆ’ax2+bxâˆ’6 is exactly divisible by x2âˆ’5x+6, then ab is:

A
an integer
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B
an irrational number
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C
611
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D
13
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Solution

## The correct option is A an integerLet the given polynomial be p(x)=x3+ax2+bx+6Given p(x) is divisible by (x–2), hence p(2)=0Put x=2 in p(x)p(2)=23+a(2)2+b(2)+6⇒0=8+4a+2b+6⇒4a+2b=–14⇒2a+b=–7 (1)It is also given that when p(x) is divided by (x–3) the remainder is 3That p(3)=3Put x=3 in p(x)p(3)=33+a(3)2+b(3)+6⇒3=27+9a+3b+6⇒9a+3b=–30⇒3a+b=–10 (2)Subtract (2) from (1)2a+b=–73a+b=–10-------------------–a=3∴a=–3Substitute a=–3 in (1), we get2(–3)+b=–7∴b=–1ab=−3−1=3

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