To find the polynomial function of the lowest degree with a leading coefficient of 1 and given roots, we need to consider the roots provided:
- The root i (which is an imaginary number)
- The root 2 (with a multiplicity of 2)
Since we are working with real polynomials, the presence of the imaginary root i implies that its complex conjugate, -i, must also be a root. Therefore, the complete list of roots we will use to construct our polynomial is:
- i
- -i
- 2 (with multiplicity 2)
Next, we can write the polynomial as a product of its factors:
The factors corresponding to the roots are:
- For root i:
(x - i) - For root -i:
(x + i) - For root 2 (with multiplicity 2):
(x - 2)^2
Now, we multiply these factors together to form the polynomial:
f(x) = (x - i)(x + i)(x - 2)^2
The product of the first two factors (x – i)(x + i) simplifies as follows:
(x - i)(x + i) = x^2 + 1
Now, we simplify the entire expression:
f(x) = (x^2 + 1)(x - 2)^2
Next, we can expand the second term:
(x - 2)^2 = x^2 - 4x + 4
Now we expand the whole expression:
f(x) = (x^2 + 1)(x^2 - 4x + 4)
Distributing (x^2 + 1) across (x^2 - 4x + 4):
f(x) = x^4 - 4x^3 + 4x^2 + x^2 - 4x + 4
Combining like terms gives:
f(x) = x^4 - 4x^3 + 5x^2 - 4x + 4
Thus, the polynomial function of lowest degree with a leading coefficient of 1 and the specified roots is:
f(x) = x^4 - 4x^3 + 5x^2 - 4x + 4