The problem involves finding the remainder when the polynomial \(x^{80} - 8x^{30} + 9x^{24} + 5x + 6\) is divided by \(x + 1\). This falls under the topic of polynomial division and the Remainder Theorem.
Participants are exploring the application of the Remainder Theorem and discussing the implications of substituting values into the polynomial. There is a productive exchange regarding the reasoning behind using \(x = -1\) and the relationship to the divisor \(x + 1\).
Some participants question the necessity of performing long division when the Remainder Theorem provides a more straightforward approach. The discussion reflects a mix of understanding and uncertainty regarding the theorem's application.
Dick said:Yeah, you would get stuck doing the division. It's a long haul. But look, suppose you did do the division f(x)=(x^80 - 8x^30 + 9x^24 + 5x + 6) by (x+1)? Then you would get f(x)=q(x)*(x+1)+r, right? Where q(x) is the quotient and r is the remainder. What happens if you put x=(-1) into that?
duki said:Thanks. How did you know to use -1?
Dick said:What's a? If a=x+1 then x=a-1. You are onto the second guess.