For each polynomial function, use the remainder theorem to find ƒ...

Question

For each polynomial function, use the remainder theorem to find ƒ(k).
ƒ(x) = x^2 - 5x+1; k = 2+i

Accepted Answer

Hey, everyone here we are asked to evaluate F of K using the remainder theorem. Here, our given function is F of X is equal to X squared minus three, X plus five. And our given value for K is four plus I. Here we have four answer choice options. Answer a zero, answer B eight plus five, I answer C eight minus five I and answer D five plus eight I. So here, before we can evaluate F of K, we need to recall the remainder theorem which states that if a polynomial F of X is divided by the expression X minus K, then we know that the remainder which I will represent as R here, the remainder R is equal to F of K. So now using this theorem here, we can find F of K. So we first take our function F of X and then we substitute in K four X which tells us we actually need to substitute in the given expression four plus I. So we need to find F of four plus I by substituting in four plus I four X from our original function. So doing this, we have the quantity of four plus I squared minus three times the quantity of four plus I plus five. So now we just want to work on simplifying this expression here of F F K. So we see F of K first, by factoring out our expression, we have F of K is equal to 16 plus A I plus I square squared minus minus three I plus five. And so now our step in simplifying is to recall that I squared is equal to negative one. So now we just want to substitute in this value negative one for wherever we have I squared in our expression. And so doing this, we now have 16 plus A I plus negative one, 12 minus three I plus five. And now our last step here is to just combine like terms. And we see that our final expression for F F K is eight plus five I. And so with this expression for F F K, we know that by the remainder theorem, it also means that this expression eight plus five I is the remainder for when F of X is divided by X minus the quantity of four plus I. And so we see here that the remainder or the expression for F F K is answer choice B eight plus five. I, thanks for watching. I hope you found this video helpful and I'll see you again next time.

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x^2 - 5x+1; k = 2+i

Key Concepts

Polynomial Functions

Remainder Theorem

Complex Numbers