Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Use the Remainder Theorem to find a remainder or test whether x-a is a factor
Problem
Use the Remainder Theorem to find the remainder when \(x^7~-~3x~+~2\) is divided by \(x~-~1\).
Big Picture
What this problem is really about
The seventh degree looks intimidating, but the divisor x minus 1 makes long division unnecessary. We’ll use the Remainder Theorem to identify the remainder as p of positive 1, substitute that value into the polynomial, and simplify with the original signs intact. That single evaluation also tells us whether the divisor is a factor.
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