Use the remainder theorem to: a.)find the remainder when f(x) is divided by x-c b.)determine whether x-c is a factor of f(x)f(x)= x^4 -5x^2 -36; x-6
Use the remainder theorem to find the remainder when `f(x)=x^4-5x^2-36` is divided by `x-6` and determine if `x-6` is a factor of `f(x)` .
The remainder theorem states that if `f(x)` is a polynomial, then the remainder when `f(x)` is divided by `x-k` is equal to `f(k)`
So we find `f(6)=6^4-5*6^2-36=1080` .
It is somewhat more convenient to use synthetic division (also known as synthetic substitution) because you not only get the remainder, you get the quotient also. Using synthetic division we get:
6 | 1 0 -5 0 -36
1 6 31 186 1080
Thus `(x^4-5x^2-36)-: (x-6)=x^3+6x^2+31x+186 +1080/(x-6)`
The factor theorem states that `x-k` is a factor of `f(x)` if `f(x) -: (x-k)` has remainder zero, or from the remainder theorem if `f(k)=0` . In this case, `f(6)!=0` so `x-6` is not a factor of `f(x)` .