Question

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

Answer 1

Answer:

The required polynomial is $P(x)=x^4-10x^3+46x^2-96x+72$.

Step-by-step explanation:

If a polynomial has degree n and $c_1,c_2,...,c_n$ are zeroes of the polynomial, then the polynomial is defined as

$P(x)=a(x-c_1)(x-c_2)...(x-x_n)$

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

$R(x)=(x-3+3i)(x-3-3i)(x-2)(x-2)$

$R(x)=((x-3)+3i)((x-3)-3i)(x-2)^2$

$R(x)=(x-3)^2-(3i)^2((x-3)-3i)(x-2)^2$     $[a^2-b^2=(a-b)(a+b)]$

$R(x)=(x^2-6x+9-9(i)^2((x-3)-3i)(x-2)^2$

$R(x)=(x^2-6x+18)(x^2-4x+4)$                $[i^2=-1]$

$R(x)=(x^2-6x+18)(x^2-4x+4)$

$R(x)=x^4-10x^3+46x^2-96x+72$

Therefore the required polynomial is $P(x)=x^4-10x^3+46x^2-96x+72$.