1976 AHSME Problems/Problem 19

A polynomial $p(x)$ has remainder three when divided by $x-1$ and remainder five when divided by $x-3$ . The remainder when $p(x)$ is divided by $(x-1)(x-3)$ is

$\textbf{(A) }x-2\qquad \textbf{(B) }x+2\qquad \textbf{(C) }2\qquad \textbf{(D) }8\qquad \textbf{(E) }15$

Solution

We know that $p(1)=3$ and $p(3)=5$ . We write $p(x)$ as $p(x)=q(x)(x-1)(x-3)+r(x)$ , where $r(x)=ax+b$ . Plugging in $x=1$ , we get $a+b=3$ . Plugging in $x=3$ , we know $3a+b=5$ . We have a systems of equations, where we can solve that $a=1$ and $b=2$ . So, our answer is $1(x)+(2)=x+2\Rightarrow \textbf{(B)}$ . ~MathJams

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1976 AHSME Problems/Problem 19

Solution