To create a polynomial equation of degree 3 with the specified roots of 2 and an imaginary number, we need to remember a couple of important principles about polynomials and their roots.
1. **Understanding Roots**: Since the coefficients of the polynomial are usually assumed to be real, complex roots come in conjugate pairs. So, if one of the roots is an imaginary number, say i (where i is the imaginary unit), the other root must be its conjugate, -i.
2. **Roots of the Polynomial**: Given these, we can say the roots of our polynomial are 2, i, and -i.
3. **Constructing the Polynomial**: A polynomial can be constructed by multiplying its factors formed by the roots. For our roots, the factors are:
- (x – 2)
- (x – i)
- (x + i)
Thus, we can write the polynomial as:
(x - 2)(x - i)(x + i)
4. **Simplifying the Polynomial**: First, we tackle the complex roots:
(x - i)(x + i) = x² - i² = x² - (-1) = x² + 1
Now we have:
(x - 2)(x² + 1)
5. **Final Expansion**: Finally, we expand this expression:
(x - 2)(x² + 1) = x(x² + 1) - 2(x² + 1) = x³ + x - 2x² - 2
Combining like terms gives us:
x³ - 2x² + x - 2
So, the polynomial equation of degree 3, having roots 2, i, and -i, is:
x³ – 2x² + x – 2 = 0
This completes the construction of the polynomial equation!