The polynomial f(x)=x² −2x³ + 3x² − ax+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 f(x) is divided by (x-2)
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Given f(x) = x4 - 2x3 + 3x2 - ax + b
When f(x) is divided by (x-1), it leaves a remainder 5
f(1) = 5
1 - 2(1)3 + 3(1)2 - a(1) + b = 5
1 - 2 + 3 - a +b = 5
-a + b = 3 … (i)
When f(x) is divided by (x+1), it leaves a remainder 19
f(-1) = 19
(-1)4 - 2(-1)3 + 3(-1)2 - a(-1) + b = 19
1 + 2 + 3 + a + b = 19
a +b = 13 … (ii)
Adding (i) and (ii),
2b = 16 b = 8
(i) a = b - 3 = 8 - 3 = 5
Therefore, f(x) = x4 - 2x3 + 3x2 - 5x + 8
When f(x) is divided by x - 3, the remainder will be
f(3) = 34 - 2(3)3 + 3(3)2 - 5(3) + 8 = 81 + 9 - 15 + 8 = 83
Step-by-step explanation:
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