The polynomial in standard form with the given solutions is x^2 −4x+4.
To write the polynomial with the given solutions in standard form, you can use the factored form of the polynomial and then expand it.
The solutions are 2 - 3i and its conjugate 2 + 3i.
The factored form of the polynomial is:
(x−(2−3i))(x−(2+3i))
Now, let's expand this expression:
(x−2+3i)(x−2−3i)
Using the distributive property, we can multiply the terms:
x(x−2−3i)−2(x−2−3i)+3i(x−2−3i)
Now, distribute the terms:
x^2 −2x−3ix−2x+4+6i+3ix−6−9i^2
Combine like terms and simplify:
x^2 −4x−5−6i+6i+9
Combine the real and imaginary parts:
x^2 −4x+4
So, the polynomial in standard form with the given solutions is x^2 −4x+4.