construct a polynomial function with the following properties: fifth degree, 2 is a zero multiplicity 3,-5 is the only other zero,leading coefficient is 3

Answer 1

Hi Carson,

Construct a polynomial function with the following properties: fifth degree, 2 is a zero multiplicity 3,-5 is the only other zero,leading coefficient is 3

Based on this information we will apply the general polynomial factor form:

y = a (x - r₁)ⁿ (x - r₂)ⁿ, where r₁ and r₂ are the zeros in factored form and n, represents the multiplicity of the factor.

For this problem, we are given a = 3, zero 2 with multiplicity of 3, and -5 which will have a multiplicity of 2 because we can add up the multiplicities to find the degree of the polynomial, thus 5- 3 = 2.

Step 1) We convert zeroes back to factor form, setting both equal to x, as follows:

x = 2 and x = -5 → At this point, we use algebra/arithmetic to set these expressions to zero getting the following: x - 2 = 0 and x + 5 = 0

We will plug this information into the above factor form and we get the following:

y = 3 (x - 2)³ (x + 5)²

And voila! we have the degree five polynomial. :)