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) = x^3 + 3x^2 -x + 1; k = 1+i

Accepted Answer

Hey, everyone in this problem, we're asked for the given polynomial function perform synthetic division to test if K equals three minus I is a zero case, it not evaluate F at K. The function F of X were given is equal to X squared minus six, X plus 10. And again, that K value is three minus I, we're given four answer choices. Option A no, it's not a zero. An F evaluated at K is three plus I. Option B is no. All right. Option C is no three and option D. Yes, it is a zero. OK? All right. So we wanna figure out if 3 -1 is a zero. So if 3 -1 is a zero, then we have a factor X -3 -1. OK. So what we wanna do is we wanna divide our function X squared minus six, X. Whoops. Not 26 minus six, X plus 10 Divided by that factor X -3 -1. So we're gonna set up our synthetic division table on the left hand side of our table. We have This zero. We're looking for 3 -1 or looking to figure out whether or not, it's a zero. OK. So we have a value of 3 - and then we're gonna start filling in our table. We're gonna start from the left with the coefficient corresponding to the highest exponent. We're gonna work our way down to the right to the coefficient with the constant term. So the highest exponent we have is X squared and so the coefficient is just one, we move to the right, our next term, OK. The next highest coefficient is one or sorry, the next highest exponent is one with a coefficient of -6. And then our constant term of 10. Now the first step is to take this -1 on the left hand side and bring it straight down and write it underneath the table. Now we're gonna take this value of one at the bottom of our table and multiply it to the value on the left of our table. OK? So we have one multiplied by three minus I, This gives us 3 -1 and we're gonna write it to the right and above. OK. So just to the right and above inside the table. Mhm Now we have a complete Callum then we're gonna add within the column. So we're adding negative six plus three minus I, This gives us -3 -1 and we write it down below. Now we have a new value down below. And so we're gonna multiply it up to the value on the left, We have negative 3 -1 Multiplied by 3 -1 yeah, I'm gonna work that out just on the side here. So we don't clutter our table. So we have three minus I multiplied by negative three minus I OK. You can use the foil method or whichever method you like to expand. And we multiply the constant terms. We get negative nine, We get negative three IK 3 multiplied by negative I and we have negative I multiplied by negative three. And this gives us plus three, I and we get a plus I squared term. So we have negative three, I plus three. I those add to zero, we're left with negative nine plus I squared. OK? Recall that I squared is defined as negative one. So we have negative nine minus one and this equals negative 10. OK? So we get the value of -10. We're gonna put that to the right and above in our table. OK? We have a complete column. So now we add within our column 10 plus negative 10 gives us zero. OK? So we're done our synthetic division. Now we need to interpret the results. OK? This final answer, the final number on the far right? And the bottom of our table is a remainder. OK? And in this case, it's equal to zero, the remainder equals zero. OK? Which tells us That K equals 3 -1 is a zero. All right. So we did our division. We found that the remainder zero, which tells us that that is in fact a zero. Ok. So we have the answer choice. D yes. Three minus. I is a zero. 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) = x^3 + 3x^2 -x + 1; k = 1+i

Key Concepts

Synthetic Division

Polynomial Functions

Complex Numbers