Mathematics[[1]] is the remainder when x3+x2+x+1 is divided by x-12  by using the remainder theorem.

[[1]] is the remainder when x3+x2+x+1 is divided by x-12  by using the remainder theorem.


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    Solution:

    In this question, we have to find the remainder when x3+x2+x+1
     is divided by x−12
     by using the remainder theorem. Before proceeding with this question, let us see what the remainder theorem is. Let us consider a polynomial f(x). When we divide f(x) by simple polynomial x – c, we get,
    f(x)=(x−c)q(x)+r(x)
    where q(x) is the quotient and r(x) is the remainder.
    Here, the degree of (x – c) is 1. So, the degree of r(x) would be 0. So, we can say that r(x) is nothing but a constant r. So, we get,
    ⇒f(x)=(x−c)q(x)+r
    Now, when we substitute x = c, we get,
    ⇒f(c)=(c−c)q(c)+r
    ⇒f(c)=0+r
    ⇒f(c)=r
    So, from this, we can say that the remainder theorem states that when we divide a polynomial f(x) by x – c, the remainder is f(c).
    Now, let us consider our question. Here, we are given a polynomial x3+x2+x+1
     so we get,
    f(x)=x3+x2+x+1......(i)
    Here, we are dividing f(x) by x−12.
     So, we get c=12.
    So, we get the remainder when f(x) is divided by x – c as
    f(c)=f(12)
    By substituting x = 12  in equation (i), we get,
    ⇒f(12)=( 12)3+(12)2+(12)+1
    ⇒f(12)= 18 + 14 + 12 +1
    ⇒f(12)=1+2+4+88
    ⇒f(12)=158
    Hence, we get the remainder as 158 when x3+x2+x+1 is divided by x− 12.
     
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