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# Function and Polynomials

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Write a Polynomial f(x) in complete factored form that satisfies the conditions: The leading coefficient is 10. The degree is 6. there is a zero at 2 with multiplicity 3, a zero at 6 with multiplicity 1 and a zero of 5i.

Guest May 1, 2017
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\(f(x)= ?\)

There is a zero at 2 with multiplicity 3

\(f(x) = (x-2)^3 (?)\)

A zero at 6 with multiplicity 1

\(f(x)= (x-2)^3(x-6)(?)\)

A zero of 5i

\(f(x)=(x-2)^3(x-6)(x^2+25)\)

It is good to note that complex factors come in pairs so by adding a complex factor you essentially solve for two complex zeroes

Now we have a polynomial with a degree of 6. However, we need the leading coefficient to be 10. To do this we just include a ten as a subset of the function like this. \(f(x)=10(x-2)^3(x-6)(x^2+25)\)  Now you have a polynomial f(x) that satisfies all the given conditions.

JonathanB  May 1, 2017
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Good answer! Though since the question asks for the complete factored form you might replace the \((x^2+25)\)  term by  \((x-5i)(x+5i)\)

Alan  May 1, 2017