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)=x^4+5x^3+5x^2−5x−6;f(3)

Accepted Answer

everyone. Welcome back in this problem we are asked to determine F of minus one. Okay, And the function F we're given is F of x equals minus three X to the four minus X cube minus five X squared plus eight X plus three. Okay. We're told to use synthetic division. And the remainder theorem. Now let's recall what the remainder theorem tells us. Okay, so very briefly the remainder theorem tells us that if we divide a function F of X, ok by some factor x minus A, then the remainder of that division is going to be F evaluated at a. Okay, Alright. So let's look at the problem we have and figure out what we want to do. Well we want to know f of -1. Okay, so we want a to B -1. We know F of X. F of X is given by minus three X. To the four minus X cubed minus five X squared plus eight X plus three. Okay, now if A is minus one then our factor x minus A will be x minus minus one. Which is going to be X plus one. Okay, so this is going to be the factor we use. So now we want to divide our function F by this factor and we're told to use synthetic division. So let's set up our synthetic division table. We're gonna fill it in. Starting with this term on the left and this is gonna be minus one. Here in this minus one is just our a value. Okay, so what we do is we take the term in the denominator which in this case is X plus one. The term. We're dividing by we set this equal to zero. Okay. And we get our a value which is the value we use here. Alright. For the rest of the top row, what we're gonna do is we're gonna fill in the coefficients corresponding with on the left, the highest exponent from our function all the way down to the constant term on the right. Okay. So we have minus three corresponding with our X to the four term. We have minus one corresponding with our X cubed term minus five corresponding with our X square term, eight corresponding with our X term. And finally three. The constant term. And to start our synthetic division we bring this -3 on the left. We bring it down and under the table. Okay And now we take what's beneath the table? We multiply it by the term on the left, so minus three times minus one. That's gonna give us three. We put it up into the right and then we add within the column minus one plus three. That's going to give us two. Now we have a new number on the bottom, multiply it by the number on the left, two times minus one gives us minus two. Okay adding in the column -5 times -2 is going to give us -7 a new number in the bottom minus seven times the minus one on the left hand side. That's going to give us seven, adding in the column eight plus seven gives us 15. Okay, multiplying one more time. This number in the bottom time is the minus one. That gives us minus 15. And adding in the column three plus minus 15 gives us minus 12. Okay, so we've done our synthetic division. Now. Let's interpret the results. When we have synthetic division. We know that this last number here the minus 12 in this case is a remainder. Let's call it R. Okay, so we know that the remainder is going to equal to F F. A. By the remainder theorem. Okay, in this case are a value is minus one, so F at minus one is going to equal the remainder minus and minus one. Well that's exactly what we were looking for. Okay, so the value F f minus one is going to be minus 12 that corresponds with answer C. I hope this video helped. Thanks everyone for watching

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