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Question

The polynomial p(x)=x42x3+3x2ax+b when divided by (x-1) and (x +1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when p(x) is divided by (x -2).

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Solution

Let:p(x)=x42x3+3x2ax+b

Now,When px is divided by x-1, the remainder is p1.When px is divided by x+1, the remainder is p-1.

Thus, we have:

p(1)=142×13+3×12a×1+b=12+3a+b=2a+b

And,

p(1)=142×13+3×12a×1+b=1+2+3+a+b=6+a+b

Now,

2a+b=5(1)

6+a+b=19(2)​​​​​​​


Adding (1) and (2), we get:

8+2b=242b=16b=8

By putting the value of b, we get the value of a, i.e., 5.

∴ a = 5 and b = 8

Now,

p(x)=x42x3+3x25x+8

Also,

When px is divided by x-2, the remainder is p(2).

Thus we have:

p(2)=242×23+3×225×2+8=1616+1210+8=10


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