Find all zeros of f(x) = x³ + 5x² – 8x + 2.

Question

Accepted Answer

Hello today we're going to be fighting the zeros of the higher order polynomial. So the polynomial given to us is X cubed plus x squared minus 15 X plus nine. Now we need to go ahead and factor this and in order to factor this to find the zeros, we can go ahead and use something called the P Q method. So the last term is going to be P and the leading coefficient is going to be cute. Now P and Q represent the factors of your last term over your first term. So P over Q is going to equal to plus or minus the factors of nine over the factors of one. While the factors of nine is going to be one, three and nine and that's all going to be over the factors of one, which is just one. So the possible zeros for this polynomial is going to be plus or minus +13 and nine. So now we just need to go ahead and pick one of the possible zeros. And synthetically divide our polynomial. And the synthetic division needs to be able to produce a remainder of zero in order for it to be a zero. So let's go ahead and pick the value negative, no positive three. So if we pick positive three, We first list out the leading coefficients of a polynomial which is going to be 1, -15 and nine. And we're going to synthetically divide these by three. And we start that process by bringing down the first coefficient which is one. Now what goes underneath the next coefficient is going to be one times the possible zero. So one times three is going to give us three and we go ahead and add these values together. So one plus three is going to give us four. Now we repeat this process until we reach the end of the synthetic division. So four times three is going to give us 12 and negative 15 plus 12 is going to give us negative three and negative three times three is going to give us negative nine and positive nine minus nine is going to give us zero. Now, since we have a remainder of zero, that means that three is going to be a zero for X. So X is equal to three is one of the zeros we're looking for now we need to go ahead and list out our new polynomial with the new leading coefficients. And when we list that out, the new polynomial is going to have one power less than the original. So instead of starting with X cubed, we're gonna start with x squared. So one x squared or x squared is gonna be our new leading polynomial or leading coefficient Plus four, X -3. And that's gonna be equal to zero. Now, unfortunately this is not a fact Herbal trin Amiel but we can go ahead and use the quadratic equation in order to find the rest of the zeros for X. So in order to use the quadratic equation, we're going to list out are giving coefficients, so A is equal to one, B is equal to four and C is equal to negative three. And we're gonna plug that into the quadratic formula. So X is equal to negative B which is negative four plus or minus. The square root of four squared minus four times A which is one times C which is negative three, all over two times A which is two times one. And now we just need to go ahead and simplify four squared is going to give us 16 and negative four times one is gonna give us negative four negative four times negative three is gonna give us positive 12. So X is equal to negative four plus or minus the square root of 16 plus 12. All over two times 1 which is two and 16 plus 12 is going to give us the value of 28. So we have X is equal to negative four plus or minus the square root of 28/2. And the square root of 28 can be rewritten as two times the square root of seven. So X is equal to negative four plus or minus two times the square root of seven. All over two. Now in the numerator we have negative four and two. So both of the terms here have a common factor of two. So we can go ahead and factor out A two from the numerator. Doing so is going to give us X is equal to two times negative two plus or minus the square root of seven. All over two. And since there is a two in both the numerator and denominator, we can go ahead and cancel them out and that's going to leave us with X is equal to negative two plus or minus the square root of seven. This cannot be simplified any further. So that means that there are the two values of X here is going to be X is equal to negative two plus the square root of seven and X is equal to negative two minus the square root of seven. And that's gonna be the remaining zeros for X. So going back to earlier three produced a zero for the remainder for the synthetic division. So one of the zeros for X is x is equal to negative three. And after further simplification through the quadratic formula, we got the other values which is X is equal to negative two plus the square root of seven and X is equal to negative two minus the square root of seven. So those are gonna be the three values of X for this given polynomial. And with that being said, the answer to this problem is going to be a So I hope this video helps you understanding how to find the zeros for this higher order polynomial and I'll go ahead and see you all in the next video

Find all zeros of f(x) = x³ + 5x² – 8x + 2.

Key Concepts

Polynomial Functions

Finding Zeros

The Rational Root Theorem