The polynomial f ( x ) = x^4 - 2x^3 +x^2 -ax + b when divided by (x - 1 ) and ( x + 1 ) leave the remainder 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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Substitute x = 1, the result should be equal to 5
1-2+1-a+b = 5
b - a = 5
Substitute x = -1, the result should be equal to 19
1+2+1+a+b = 19
a+b = 15
Adding both, b = 10 and a = 5
So, the polynomial is x^4-2x^3+x^2-5x+10 = 0
Dividing it with x-2,
Quotient is x²+x-3
Remainder is 4
1-2+1-a+b = 5
b - a = 5
Substitute x = -1, the result should be equal to 19
1+2+1+a+b = 19
a+b = 15
Adding both, b = 10 and a = 5
So, the polynomial is x^4-2x^3+x^2-5x+10 = 0
Dividing it with x-2,
Quotient is x²+x-3
Remainder is 4
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