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Question

In the following cases, use the remainder theorem and find the remainder when p(x) is divided by g(x).
p(x)=x3+3x25x+8 g(x)=x3

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Solution

The Remainder Theorem states that when we divide a polynomial p(x) by any factor (xa); which is not necessarily a factor of the polynomial; we will obtain a new smaller polynomial and a remainder, and this remainder is the value of p(x) at x=a, that is p(a).

Here, it is given that the polynomial p(x)=x3+3x25x+8 and the factor is g(x)=x3, therefore, by remainder theorem, the remainder is p(3) that is:

p(3)=33+(3×32)(5×3)+8=27+(3×9)15+8=27+2715+8=6215=47

Hence, the remainder is r(x)=p(3)=47.


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