To find the polynomial function of the lowest degree with a leading coefficient of 1 and the given roots, we start by writing down the roots. We have the roots i (which is a complex root) and 2 (which is a real root, with a multiplicity of 2).
For complex roots, we know that if a polynomial has real coefficients, then the complex roots must appear in conjugate pairs. Therefore, along with the root i, the root -i must also be included.
The polynomial can be constructed from its roots as follows:
- For the root 2 with multiplicity 2, the factor is (x – 2)².
- For the root i, the factor is (x – i), and for the root -i, the factor is (x + i).
Thus, we can express the polynomial as:
(x – 2)² * (x – i) * (x + i)
Now we can simplify the expression:
The factors involving i can be simplified using the identity (x – i)(x + i) = x² + 1:
(x – 2)² * (x² + 1)
Next, we expand (x – 2)²:
(x – 2)² = x² – 4x + 4
Now we multiply this by (x² + 1):
(x² – 4x + 4)(x² + 1) = x²(x² + 1) – 4x(x² + 1) + 4(x² + 1)
This simplifies to:
x^4 - 4x^3 + 4x² + x² - 4x + 4 = x^4 - 4x^3 + 5x² - 4x + 4Thus, the polynomial function of the lowest degree with a leading coefficient of 1 and roots i, 2 (with multiplicity 2) is:
f(x) = x^4 - 4x^3 + 5x² - 4x + 4