Find a rational zero of the polynomial function and use it...

Find a rational zero of the polynomial function and use it to find all the zeros of the function. 25) f(x) = 3x^3 - 17x^2 + 18x + 8 26) f(x) = 3x^3 - x^2 - 18x + 6 27) f(x) = x^4 + 4x^3 - 11x^2 - 26x - 12 28) f(x) = 3x^4 + 29x^3 + 111x^2 + 179x + 78

Transcript

00:01 All right, so for problem 741, we're given this function, and we have to use the rational zero theorem to find its zeros, or basically when this function is equal to zero.

00:13 Now, the rational zero theorem is kind of hard to explain, so we're just going to go ahead and deal with it.

00:19 So first, we find a constant term, which is the term without the x.

00:25 So it's going to be this 8, and we're just going to record it here.

00:29 And then we find the term with the highest x -molent, which is this x -to -the -fifth.

00:36 And we're going to record it here.

00:39 And you can imagine this as 1 times x -to -the -fif.

00:43 But really, all we care about is the coefficient.

00:47 So we can just erase this x -to -the -fifth and be left with the 1.

00:52 And now for both numbers, we find all their factors.

00:55 So for 8, it's going to be 1, 2, 4, and 8.

01:00 And we're going to set these equal to the p values.

01:05 And for all the factors of one, well, it's just going to be one.

01:09 And we're going to set that equal to the q value.

01:12 And now we divide each p value by each q value.

01:16 Fortunately, there's only one q value.

01:19 So we just divide all the p values by one and that's it.

01:22 So it's just going to give us 1, 2, 4, and 8.

01:28 And thus our full list of 4.

01:31 Possible zeros is going to be positive, come on, come on, positive or negative, 1, 2, 4, and 8...

Question

Find a rational zero of the polynomial function and use it to find all the zeros of the function. 25) f(x) = 3x^3 - 17x^2 + 18x + 8 26) f(x) = 3x^3 - x^2 - 18x + 6 27) f(x) = x^4 + 4x^3 - 11x^2 - 26x - 12 28) f(x) = 3x^4 + 29x^3 + 111x^2 + 179x + 78

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