Finding a polynomial with intereger coefficients that satisfies the given conditions: p(x)=3, and zeros 3, 2i, and -2i. I can not figure how to solve this, Please Help!
To find a polynomial with integer coefficients that satisfies the given conditions, namely p(x) = 3 and zeros 3, 2i, and -2i, you can use the fact that complex roots occur in conjugate pairs.
First, let's consider the complex roots: 2i and -2i. Since complex roots occur in conjugate pairs, the corresponding factors of the polynomial will be (x - 2i) and (x + 2i).
Next, let's consider the real root: 3. The corresponding factor of the polynomial will be (x - 3).
Now, multiply these factors together to obtain the polynomial:
p(x) = (x - 2i)(x + 2i)(x - 3)
To simplify this expression, recall that (a - b)(a + b) = a^2 - b^2. Applying this formula, we can rewrite our polynomial as:
p(x) = (x^2 - (2i)^2)(x - 3)
= (x^2 + 4)(x - 3)
Finally, expand this expression to get the polynomial with integer coefficients:
p(x) = x^3 - 3x^2 + 4x - 12
Therefore, a polynomial satisfying the given conditions is p(x) = x^3 - 3x^2 + 4x - 12.