Find the value of a and b so that (x-1) and (x-2) are factors of x^3+ax^2+2x+b.
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since (x-1) and (x-2) are factors of the given polynomial, for x=1,2 the polynomial must be equal to 0.
So, for x=1, 1³+a×1²+2×1+b=0
=>1+a+2+b=0
=>a+b+3=0----------(i)
for x=2, 2³+2²a+2×2+b=0
=>8+4+4a+b=0
=>12+4a+b=0---------(ii)
(ii)-(i)=>3a+9=0
=>a=-3
let's put a's value in equation (i).
-3+b+3=0
=>b=0
So, for x=1, 1³+a×1²+2×1+b=0
=>1+a+2+b=0
=>a+b+3=0----------(i)
for x=2, 2³+2²a+2×2+b=0
=>8+4+4a+b=0
=>12+4a+b=0---------(ii)
(ii)-(i)=>3a+9=0
=>a=-3
let's put a's value in equation (i).
-3+b+3=0
=>b=0
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