Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 4, -8, and 2 + 3i.

Answer 1

Answer:

The polynomial function is $P(x) = x^4-35x^2+180x-416.$

Step-by-step explanation:

A polynomial function is completely determined by its roots, up to a constant factor. So, if we want that $x_1=4$, $x_2=-8$ and $x_3=2+3i$ be roots of the polynomial P, we can write it as

$P(x)= (x-x_1)(x-x_2)(x-x_3)=(x-4)(x+8)(x-(2+3i)) = (x^2+4x-32)(x-(2+3i)).$

Now, notice that the factor $x^2+4x-32$ has real coefficients, while the other one don't. So, we need to ‘‘eliminate’’ the complex coefficients that will appear.  This can be done adding other complex root to the polynomial: the conjugate of $x_3$: 2-3i. Then,

$P(x) = (x^2+4x-32)(x-(2+3i))(x-(2-3i)).$

Expanding the above expression we obtain the desired polynomial

$P(x) = x^4-35x^2+180x-416.$

Recall that P(x) must have three roots, and this implies that P has at least degree 3. As we had to add a new root in order to obtain real coefficients, the degree of P must be at least 4. With this reasoning we assure the minimal degree.