Find an nth degree polynomial function with real...

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros. Do not sumplify, leave your function in factored form n=3 4 and 2 i are zeros: a_(n)=1 Select the correct choice below. a. f(x)=1(x-4)(x 2i)(x-2i) b. f(x)=1(x 4)(x 2i)(x-2i) c. f(x)=1(x-4)(x 2i) d. f(x)=1(x-4)(x-2i)(x-2i)M

Transcript

00:01 Okay, here we've got, we need to find an nth degree polynomial function with real coefficients, and we're given some conditions.

00:09 One of our conditions is that n is equal to three, so that means we've got a third degree polynomial, and we're given two of our zeros.

00:18 We have zeros of two and five i.

00:24 So this is an imaginary zero, of course, complex, and those always come in common.

00:31 So this actually lets us know our third zero.

00:37 Again, if we have a third degree polynomial, that means that we have at most three zeros.

00:43 And because complex zeros always come in conjugate pairs, this means we definitely have negative 5i as another 0.

00:51 So this gives us three zeros.

00:53 One, two, three.

00:55 So all we have to do now is set up our factors and then expand it out.

01:00 So the factors would be x minus 2, x minus 5i, and x minus a negative 5i, which is going to give us x plus 5i.

01:16 So x minus 2, x minus 5i, and x plus 5i.

01:25 Now, when you're expanding these out, i do notice that your question says if you're using a graphing utility, which means to me that you are allowed to use that.

01:35 So you can use your graphing calculator and one of your algebra functions to expand.

01:41 But if you're doing it by hand, you always want to start with your two conjugate factors and expand those out first by foiling.

01:53 So i'm going to have the x minus 2 over here.

01:55 I'm not going to do anything with that just yet.

01:57 And then here i'm going to have x times x, which is x squared, x times positive 5i x times negative 5 i which is going to give me minus x 5 i and then minus 25 i squared so what you can see here that what's going to happen is these two terms you're going to cancel each other out i squared is equal to negative 1 so that's just going to change the sign so we're going to get x minus 2 x squared plus 25.

02:38 Now i can foil this out again.

02:41 I'm going to get x cubed minus 2x squared plus 25x minus 50.

02:50 So there's my function.

02:53 There's my third degree function that has the zeros that we were provided with plus the conjugate pair.

03:02 So, okay, we have another one here.

03:14 We have n equal three...

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