Find all of the zeros of the following polynomial functions...

Find all of the zeros of the following polynomial functions by the method outlined. i) list all possible rational zeros ii) From part (i) identify a rational zero iii) Use synthetic division to find factors of the ploynomial iv) Then factor the polynomial completely to find remaining zeros a) f(x) = x^4 - 3x^3 + x^2 + 3x - 2 b) g(x) = x^4 - 4x^3 - x^2 + 14x + 10

Transcript

00:01 So for this problem we are being told to list all the possible zeros, then we're going to identify a rational zero which we can use with synthetic division to find all the factors of the polynomial.

00:14 So how do you find all possible zeros? so in a polynomial you have f of x equals and then if you're writing it in standard form that first term must have a coefficient.

00:31 If it doesn't have it's implied that it's one and then you have the last term which i'm going to use actually a p and this last term doesn't have an x and if it does it's an x to the zero power which we don't write.

00:53 So how do we find the possible rational zeros? well we divide p by q.

01:02 We divide p by q.

01:03 In this case q is one so we don't have to worry about all the different combinations that can come out when we divide.

01:11 So when we divide this we would get negative two and we also have to consider all the factors that we would have here.

01:22 So negative two, two, negative one, one.

01:27 And how do you know which one we should choose? well the first thing that you can do is that you can just do trial and error and do synthetic division with each of these and if you have a remainder of zero then it works.

01:43 If you have a remainder other than zero it doesn't work.

01:47 However sometimes you're going to have a list of a lot of numbers and doing synthetic division on each of those is going to be a headache.

01:55 So what's the shortcut? well remember that if you are finding the zeros that means that your y is going to be equal to zero.

02:07 So that means that if we were to plug in our x value here we should get zero because that would imply that we have found an x intercept.

02:23 So let's go ahead and give this a try.

02:26 Let's try with f of negative one.

02:31 So if we were to plug in f of negative one we have negative one to the fourth minus three to negative one to the third plus negative one squared plus three times negative one minus two.

02:46 So we have one plus or this actually it's not plus negative one to the third is negative one and then we have oh we do have plus three plus one minus three minus two.

03:06 So three is cancelled, one plus one is two, minus two equals zero...

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