Use synthetic division to determine whether the given number k is...

Question

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x^4 + 2x^3 -x+3; k=2/5

Accepted Answer

Hey, everyone in this video, we're gonna be working through the following problem For the given polynomial function, perform synthetic division to test if K equals 3/ is a zero. In case it's not evaluate F at K, the function F for given F of X is equal to four, X to the exponent four plus four, X cubed minus five X plus one. Again, we have that K value of 3/4. We're given four answer choices. Option A no, it's not A zero. An F at K is 13 divided by 64. Option B no, 64 divided by 13. Option C no -17, divided by 16. And option D is, yes, it is a zero. So we have our K value. We want to figure out if K equals 3/4 as a zero. If K equals 3/4 is a zero, then we get the factor X minus three quarters. OK. So we wanna take Are polynomial four, X to the exponent four was for execute minus five X plus one, Right? And we wanna divide it by that factor X -3/4. Now I'm gonna make an adjustment in the numerator. And when we do our synthetic division, we have to include every single coefficient for every single exponent from the constant term up to our highest exponent. OK. So we have our highest exponent of four. So we have an X to the four term, we have an X cube term, but we're missing the X squared term. So we're gonna write rewrite this as four X to the exponent four plus four X Q plus zero X squared minus five X plus one. OK. So we're just writing in that zero X square, that's implied because we don't wanna miss it in our synthetic division table. And again, we're dividing by X -3/4. Now we're gonna go ahead and draw our synthetic division table On the far left of our table. We wanna write the zero of the denominator. Now, the zero of the denominator is the K value we were given of 3/4, three divided by four because we wanted to figure out if that is a zero, that's the value we're testing, that's what we're dividing by in our synthetic division. Now, inside the table on the top row, we're gonna write the coefficients corresponding to starting on the left hand side, the highest exponent term, all the way down on the right hand side to the constant term. and so our highest exponent is four, The coefficient on the X to the exponent four term is four. Next, we have X cubed, the coefficient is four again. And we have our x squared term of the coefficient of zero. We have our X term with a coefficient of negative five and our constant term of one. OK. So we've got our table set up. Now we're gonna do the synthetic division. We're gonna start on the left hand side by bringing this four down and underneath our table. When we have a value underneath our table, we take it and we multiply it to the value on the left outside of our table. So we have four multiplied by 3/4 which gives us a value of three. We're gonna write that to the right and above. Now we have a complete column in our table and we're gonna add the values in that coal. We get four plus three, which gives us a value of seven. Now we have a new Value underneath their table of seven. We're gonna multiply that to the number on the left. So seven multiplied by 3/4 And we get 21 divided by four. Now we have another complete column, we add in our column zero plus 21, divided by four, Gives us 21 divided by four. We write that below Again. That's a new value underneath our table. We multiply it to our 3/ 21 divided by four, multiplied by three, divided by four, Gives us 63 divided by 16. We have a completed column So we're going to add, we wanna add negative five plus 63 divided by 16. So we're just gonna do that on the side. So we don't make everything messy inside of our table. So we have negative five plus 63 divided by 16. HM This is equal to negative 80 divided by Plus 63 divided by 16 K finding a common denominator there. And this gives us -17/16. Yeah. And so the Value of that column - divided by 16, we put that down below. And now we need to multiply again by three, divided by four. This gives us a value of -51 divided by 64. And one last time we add in this column negative divided by 64 plus one. And we get 13 divided by 64. OK? Now we're left to interpret the results. The final number on the right hand side is going to be a remainder. OK? So 13 divided by 64 is a remainder and you'll notice that that remainder is not equal to zero. OK? Since the remainder does not equal, zero OK. That means that the value of three quarters that we were investigating is not a zero. OK? So it's not a zero. And what we were asked to do next is to evaluate F K. No, we don't even need to do anything else. OK? We can use remainder theorem. And the work we've already done to do that for us. OK. So our remainder is equal to divided by 64, which tells us by the remainder theorem. The F at 3/ that value we were looking at Is equal to 13 divided by 64. It's equal to the remainder when we do that division. And so using synthetic division, we found that K equals three quarters is not a zero of the function we were given. OK. And when we evaluate the function at three quarters, we get 13 divided by 64 which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x^4 + 2x^3 -x+3; k=2/5

Key Concepts

Synthetic Division

Polynomial Function

Zero of a Polynomial