Q:

Find a polynomial with integer coefficients that satisfies the given conditions. R has degree 4 and zeros 3 − 3i and 2, with 2 a zero of multiplicity 2

Accepted Solution

A:
Answer:The required polynomial is [tex]P(x)=x^4-10x^3+46x^2-96x+72[/tex].Step-by-step explanation:If a polynomial has degree n and [tex]c_1,c_2,...,c_n[/tex] are zeroes of the polynomial, then the polynomial is defined as[tex]P(x)=a(x-c_1)(x-c_2)...(x-x_n)[/tex]It is given that the polynomial R has degree 4 and zeros 3 − 3i and 2. The multiplicity of zero 2 is 2.According to complex conjugate theorem, if a+ib is zero of a polynomial, then its conjugate a-ib is also a zero of that polynomial.Since 3-3i is zero, therefore 3+3i is also a zero.Total zeroes of the polynomial are 4, i.e., 3-3i, 3_3i, 2,2. Let a=1, So, the required polynomial is[tex]R(x)=(x-3+3i)(x-3-3i)(x-2)(x-2)[/tex][tex]R(x)=((x-3)+3i)((x-3)-3i)(x-2)^2[/tex][tex]R(x)=(x-3)^2-(3i)^2((x-3)-3i)(x-2)^2[/tex]     [tex][a^2-b^2=(a-b)(a+b)][/tex][tex]R(x)=(x^2-6x+9-9(i)^2((x-3)-3i)(x-2)^2[/tex][tex]R(x)=(x^2-6x+18)(x^2-4x+4)[/tex]                [tex][i^2=-1][/tex][tex]R(x)=(x^2-6x+18)(x^2-4x+4)[/tex][tex]R(x)=x^4-10x^3+46x^2-96x+72[/tex]Therefore the required polynomial is [tex]P(x)=x^4-10x^3+46x^2-96x+72[/tex].