Answer:
f(x) = (x+2)(x+3)(x-(3-6i))(x-(3+6i))
f(x) = 270 + 189 x + 21 x^2 - x^3 + x^4
Step-by-step explanation:
First of all, we must know that complex roots come in conjugate pairs.
So the zeros of your equation would be
x = -2
x = -3
x = 3 - 6i
x = 3 + 6i
Your polynomial is of fourth degree.
f(x) = (x-(-2))(x-(-3))(x-(3-6i))(x-(3+6i))
f(x) = (x+2)(x+3)(x-(3-6i))(x-(3+6i))
Please , see attached image below for full expression
f(x) = 270 + 189 x + 21 x^2 - x^3 + x^4
Answer:
The required polynomial is [tex]P(x)=a\left(x^4-x^3+21x^2+189x+270\right)[/tex].
Step-by-step explanation:
The general form of a polynomial is
[tex]P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}[/tex]
where, a is a constant, [tex]c_1,c_2,..c_n[/tex] are zeroes with multiplicity [tex]m_1,m_2,..m_n[/tex] respectively.
It is given that –2, –3,3 – 6i are three zeroes of a polynomial.
According to complex conjugate root theorem, if a+ib is a zero of a polynomial, then a-ib is also the zero of that polynomial.
3 – 6i is a zero. By using complex conjugate root theorem 3+6i is also a zero.
The required polynomial is
[tex]P(x)=a(x-(-2))(x-(-3))(x-(3-6i))(x-(3+6i))[/tex]
[tex]P(x)=a(x+2)(x+3)(x-3+6i)(x-3-6i)[/tex]
[tex]P(x)=a\left(x^2+5x+6\right)\left(x-3+6i\right)\left(x-3-6i\right)[/tex]
On further simplification, we get
[tex]P(x)=a\left(x^3+6ix^2+2x^2+30ix-9x+36i-18\right)\left(x-3-6i\right)[/tex]
[tex]P(x)=a\left(x^4-x^3+21x^2+189x+270\right)[/tex]
Therefore the required polynomial is [tex]P(x)=a\left(x^4-x^3+21x^2+189x+270\right)[/tex].
