Two of the roots of the equation f(x)=0 are -1 and 2. Given that x2−3x+1 is a quadratic
factor of f(x), find the remainder when f(x) is divided by 2x−1.
Since degree of f(x) is 4 and x2−3x+1 is a factor of f(x), it can be written as product of two quadratics
Therefore, f(x)=(x2−3x+1)(ax2+bx+c). Again since, 2x4 is the leading term a, must be equal to 2
So, f(x)=(x2−3x+1)(2x2+bx+c). Given −1 and 2 are the roots of f(x) ⇒ f(−1)=0 & f(2)=0
Note that x2−3x+1 does not vanishes at x=−1,2 ⇒ 2x2+bx+c must vanishes at this two points.
⇒2−b+c=0,8+2b+c=0 Solving the equations we get b=−2,c=−4
Thus f(x)=(x2−3x+1)(2x2−2x−4). Required remaninder is f(12)=98
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