Question: When a polynomial P(x) is divided by (x+2)(x−4) the quotient is the polynomial Q(x) and the remainder is ax+b. Find a,b if P(−2)=3 and P(4)=2.
Solution.
Using the factor theorem we get two equations for a,b which we can solve to find a,b.
We have P(x)=(x+2)(x−4)Q(x)+ax+b
Therefore
P(−2)=3=−2a+b⟹−2a+b=3 (1)
and
P(4)=2=4a+b⟹4a+b=2 (2)
Subtract (2)-(2) gives
−1=6a⟹a=−16
Then b=3+2(−1/6)=223=8/3.
Conclude: a=−16 and b=83.
Remember to check my working and tell me if I made an error!! (reward!)
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