In Exercises 33–40, use synthetic division and the Remainder Theo...

Question

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x^4+10x^3+5x^2+x+1;f(− 2/3)

Accepted Answer

Welcome back. I am so glad you're here. We are to determine F of negative one third. When we are given that F of X equals three X to the fourth minus two X to the third minus 10 X squared plus six X plus two. And instead of just plugging in negative one third in everywhere there's an X. We're going to use synthetic division. And the remainder theorem, we recall from previous lessons that the remainder theorem tells us that when we use synthetic division, our remainder is going to be the result of F of that value that we're using for synthetic division. So in this case F of negative one third is going to be equal to our remainder. So let's do the synthetic division and see what our remainder is. Okay, we're going to write down all the coefficients and the constant term from our F of X. So the first coefficient is three. Then we have negative two negative 10 positive six. And two. Step one in our synthetic division. And this step one only occurs the one time we're going to bring down this first term, so bring down the three. And then we've got a step one that repeats. And step one here is to multiply, we're going to take this negative one third, times three and place it up here negative one third, times three that's negative one. Step two is to add we're going to take negative two plus negative one, that is negative three. Now we go back to multiplying negative one third times negative three. That is a positive one. Back to adding negative 10 plus one. That's negative nine. Back to multiplying negative one third times negative nine. That is a positive three. Back to adding six plus three is nine. Multiplying again negative one third times nine gives us a negative three and back to adding two plus negative three gives us a negative one. There is our remainder, so F of negative one third is equal to negative one. Thank you to the remainder theorem. Alright, We look at our answer choices and that matches with answer choice. A Well done. We'll catch you on the next one.

In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x^4+10x^3+5x^2+x+1;f(− 2/3)

Key Concepts

Synthetic Division

Remainder Theorem

Polynomial Evaluation