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.
The Remainder Theorem states that when a polynomial function is divided by a linear factor, the remainder is equal to the value of the function at the root of the linear factor. In other words, if we divide a polynomial f(x) by (x-a), the remainder will be equal to f(a).
To use the Remainder Theorem, first identify the linear factor that the polynomial is being divided by. Then, plug the value of this factor into the original polynomial to find the remainder. This remainder will be the value of the function at the root of the linear factor.
Dividing a polynomial by a linear factor can help us simplify complex polynomial expressions and identify important characteristics of the function, such as its roots and end behavior.
To apply the Remainder Theorem to this polynomial, we first need to identify the linear factor, which is (x+1). Then, we can substitute -1 (the root of the linear factor) into the polynomial to find the remainder. In this case, the remainder will be -9.
Yes, the Remainder Theorem can be used for any polynomial division problem as long as the divisor is a linear factor. However, it is important to note that the remainder will only be accurate if the polynomial is divided by its simplest form (all terms are in descending order and all exponents are present).