Find the polynomial having the following roots with multiplicity

x=-3 multiplicity 3 x=3i x=3 multiplicity of 2

Find the polynomial having the following roots with multiplicity

x=-3 multiplicity 3 x=3i x=3 multiplicity of 2

Tutors, sign in to answer this question.

Hi, if x=3 is a root then (x-3) is a factor and since the multiplicity is 2, (x-3)^2 is a factor.

Next, since -3 is a root, then (x- (-3)) is a factor, that is, (x+3) is a factor, and in fact, (x+3)^3 is a factor since the multiplicity is 3.

Now, what about x=3i being a root? It means (x-3i) is a factor. But for our polynomial to have real coefficients, the "complex conjugate" of 3i must be a root as well, The complex conjugate of a+bi is a-bi. So the complex conjugate of 3i is just -3k (since a=0 here)

So both (x-3i) and (x+3i) are factors; let's multiply them and see what we get: we get,

(x-3i)(x+3i) = x^2 -3ix + 3ix -9i^2 = x^2 +9

So our polynomial has (x^2 + 9) as a factor and the smallest (lower degree) real polynomial having all the desired roots and multiplicities is:

** p(x) = (x+3)^3 * (x-3)^2 * (x^2 + 9)

where the * symbol just stands for "times" to clarify we are multiplying these (not needed mathematically but when reading computer format it helps to clarify for the reading eyes!) I suspect they want yo to leave the polynomial factored like this. Otherwise, you'll need to multiply all these together. Notice that (x+3)(x-3) = (x^2 - 9) so

(x+3)^3 * (x-3)^2 * (x^2 + 9) = (x+3)*[(x+3)^2*(x-3)^2)]*(x^2+9)

which is (x+3) * [(x^2 - 9)(x^2 - 9)]* (x^2+9)

re-grouping: (x+3) * (x^2 - 9) * [(x^2 - 9)* (x^2 + 9)]

(x+3)(x^2 - 9) * (x^4 - 81) which I leave to you to multiply out to get a 7th degree polynomial..unless they are ok with it being factored in which again, use the form ** above. Hope this helps! Let me know if you have any questions.

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.