Problem

Find the remainder when the polynomial 3x3−5x2+2x−7 is divided by the binomial x−2.

Answer

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Answer

Since x−2=0 when x=2, the remainder R(x)=R(2)=1−2Q(2). But the remainder must be a constant, so Q(2)=0 and R(2)=1.

Steps

Step 1 ：3x3−5x2+2x−7=(x−2)Q(x)+R(x), where Q(x) is the quotient and R(x) is the remainder.

Step 2 ：Substitute the value of x=2 into the polynomial equation: \ 3(2)3−5(2)2+2(2)−7=2Q(2)+R(2). \ Simplify to get: 24−20+4−7=2Q(2)+R(2), or 1=2Q(2)+R(2).

Step 3 ：Since x−2=0 when x=2, the remainder R(x)=R(2)=1−2Q(2). But the remainder must be a constant, so Q(2)=0 and R(2)=1.

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