Write a polynomial equation with integral coefficients that has 1/2 and 2i as solutions
since we have real coefficients, and 2i is a root, so is -2i. If 1/2 is a root, one factor of p(x) is (2x-1). So,
p(x) = (2x-1)(x-2i)(x+2i)
= (2x-1)(x^2+4)
To write a polynomial equation with integral coefficients that has 1/2 and 2i as solutions, we need to remember that complex roots always come in conjugate pairs.
First, let's start by considering the factors based on the given solutions:
- For 1/2 to be a solution, the polynomial should have a factor of (2x - 1).
- For 2i to be a solution, the polynomial should have a factor of (x - 2i).
Since complex roots come in conjugate pairs, the conjugate of 2i is -2i. Therefore, the polynomial should also have a factor of (x + 2i).
Now, we can multiply these factors together to get the polynomial equation:
(2x - 1)(x - 2i)(x + 2i)
To simplify, we can use the difference of squares to cancel out the imaginary terms:
(2x - 1)(x^2 + 4)
Expanding this expression, we get:
2x^3 - 8x + x^2 - 4
Lastly, we can combine like terms to write the polynomial equation with integral coefficients:
2x^3 + x^2 - 8x - 4