Math CentralQuandaries & Queries


Question from Pasandi:

f(x) is a quadratic polynomial. when f(x) is divided by
(x-1),(x-2) & (x+2) the remainders respectively are -1, 4 and 2 how to find the f(x) in a question like this?


The key here is the Polynomial Remainder Theorem which states that if a a polynomial $f(x)$ produces a remainder of $R$ when divided by $(x-k)$ then $f(k) = R.$

Suppose your quadratic polynomial is $f(x) = a x^2 + b x + c.$ Since its remainder upon dividing by $x-1$ is $-1$ the Remainder Theorem tells us that $f(1) = -1$ and hence

\[f(1) = a \times 1^2 + b \times 1 + c = a + b + c = -1.\]

In a similar fashion division by $x - 2$ and $x + 2$ give us two further linear equations in $a, b$ and $c.$ Solve for $a, b$ and $c.$


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