The remainder after division

Task number: 2631

Show that if a polynomial \(p(x)\) is divided by the linear polynomial \(x-c\), then the remainder is equal to \(p(c)\).

  • Resolution

    The division rule is \(p(x)=q(x)(x-c)+r(x)\).

    The degree of \(r\) is smaller than the degree of the linear polynomial \((x-c)\), i.e. it is a constant factor \(r(x)=r\).

    If we substitute \(x=c\) into \(p(x)=q(x)(x-c)+r\) we get \(p(c)=q(c)(c-c)+r=r\), as it was required.

Difficulty level: Easy task (using definitions and simple reasoning)
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