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 +7x^2 + 10x; k=0

Accepted Answer

Hey, everyone in this problem for the given polynomial function, we're asked to perform synthetic division to test if K equals zero is a zero. Um If it's not, we're asked to evaluate fa K, the function F of X were given is equal to X cubed plus 14 X squared plus 33 X. And we're given four answer choices. A no 84. Option B no 308. Option C no undefined and option D Yes. Now one more wanna test if K equals zero is a zero. OK? If K equals zero is a zero, then the X minus zero is going to be a factor. So what we wanna do is we want to take our function F of X X cubed plus 14 X squared plus 33 X And divide it by the factor X zero. Now, I'm gonna add on to the end of the function in our enumerator A plus zero. When we do our synthetic division, it's important to include every single term. OK. The coefficient for every single term from the highest exponent down to our constant term. So we'll write it there so that we don't forget. Now if we look at this function, hey, we wanna know if K equals zero is a zero, we can think if we were to plug zero into this equation, we get zero plus zero plus zero, we get zero. So we expect that this is going to be a zero. But let's use our synthetic division to double check. We're gonna set up our synthetic division table and we're gonna start on the left hand side with our K value. Our K value is zero. Now, the top row of our table is gonna be filled out with the coefficients corresponding to the numerator. OK. So they're gonna correspond to the function F of X. On the left hand side, we're gonna start with the coefficient of the highest exponent term. And we're gonna work our way down to the coefficient of the constant term on the right hand side. So our highest exponent term is execute has a coefficient of one. So that's what we write on the left side of that first row. Our next term x squared has a coefficient of 14. Then we have our x term with a coefficient of 33. And finally our constant term which is zero. OK. And it is very, very important to include this constant term even though it's zero. Now, in order to do our synthetic division, we start on the left hand side and we're gonna bring this one down and underneath our table When we have a value underneath our table. We're gonna multiply it to the value on the left outside of our table. So we have one multiplied by zero and this gives us zero. And we're gonna write it in the next column over in the second row of our table. Now we have a complete column. So we're gonna add within that column. 14 plus zero gives us 14. We write it down below. We have a new number below our table. We're gonna multiply it to the number on the left. 14 multiplied by zero gives us zero. Yeah, we write that in the next column over. Now we have a complete column. We add within that column. We get 33 plus zero. This gives us 33. OK. That's a new number below. We multiply it to our K value on the left. 33 multiplied by zero gives us zero. And then we add this last complete column zero plus zero gives us zero. Now this last value on the right hand side underneath our table is our remainder. And in this case, our remainder Is equal to zero. If the remainder is equal to zero, that tells us that K equals zero is a zero. All right. So we did our synthetic division, we found a remainder of zero which tells us that K equals zero is in fact a zero of our function F of X. And so the correct answer choice here is option D Yes, 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 +7x^2 + 10x; k=0

Key Concepts

Synthetic Division

Polynomial Function

Zero of a Polynomial