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)=2x^4−5x^3−x^2+3x+2;f(− 1/2)

Accepted Answer

Hey everyone in this video we are asked to determine the value of the function F evaluated at minus 3/4 were given the function F. Of X equals four X. To the four minus X cubed minus seven X squared plus five X. We're told to use synthetic division and the remainder theorem. Alright. So for a quick recall, the remainder theorem, short form here tells us that if we divide a polynomial F of X OK by a factor X minus a, then the remainder of that division will be F evaluated at it. Okay. Alright. So that's just a quick refresher on the remainder theorem. And now let's go ahead. So in this case we have F of X. The function we're given and I didn't this plus two is on there too. So we have F of X is equal to four X. To the four minus X cubed minus seven X squared plus five X plus two. Okay, that's our f. Now if we're looking to find F at minus three, force we're going to say that A. Is minus 3/4. Okay. And then our factor is going to be X minus minus 3/4, which is just X plus 3/4. Alright, so the synthetic division, the division we want to do is going to be or X to the fore minus X cubed minus seven X squared plus five X plus two. All divided by X plus 3/4. Alright, so let's set up the synthetic division table. Okay. The first number we're gonna fill in is the number on the left hand side here. That's just gonna be our value of a. Okay so we want the root of this denominator which turns out to be a in these cases. Alright, filling in the rest of the top row. We're going to use the coefficients of our function F. We're gonna take from the highest exponent on the left, all the way down to the lowest exponent on the right. Okay so the coefficient the first coefficient is four corresponding with the highest exponent of four. Then we have minus one corresponding with the coefficient of three. Or sorry with the exponent of three. We have minus seven corresponding with the X squared term, five corresponding with the X term. And finally to the constant term. Alright. In order to do our synthetic division, we start on the left hand side, we bring that four all the way down into the bottom. Okay and now we multiply what's on the bottom by what's on the left? Four times minus three. Over four is going to be minus three. Okay we add in the column minus one plus minus three is minus four. Again we have a number on the bottom, we multiply it by the number on the left minus four times minus 3/4. Well that's going to give us positive three, Adding in the column -7-plus 3 is gonna give us - again. Alright so again multiplying minus four by minus 3/4. We get three. Again adding in the column five plus three gives us eight number on the bottom times the number on the left. Eight times minus 3/4. That's going to give us minus six and finally adding in the column to plus minus six, that's going to give us minus four. Okay? Now in order to interpret our results, Well this last number here, this is our remainder, we can call it our Okay well that's what we're looking for. Okay, so the remainder we know from the remainder theorem is going to be F. At A Which in this case is F at -3/ which is equal to -4. That's the value we were looking for. Okay now we can double check this, we want to double check, we can verify by plugging in -3/4 into our function, working that out and we can verify that this is indeed the value we were looking for. Okay, so that's going to correspond with solution D. I hope this helps everyone see you in the next video

Struggling with College Algebra?

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

Dividing Polynomials practice set