Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

5, -3, and -1 + 3i

f(x) = x4 + 12.5x2 - 50x - 150
f(x) = x4 - 4x3 + 15x2 + 25x + 150
f(x) = x4 - 4x3 - 15x2 - 25x - 150
f(x) = x4 - 9x2 - 50x - 150

Answer 1

`f(x)=x^4-9x^2-50x-150`

Explanation:

Let f(x) be the polynomial function of minimum degree with real coefficients whose zeros are 5, -3, and -1 + 3i be f(x).

By the complex conjugate property of polynomials, -1-3i is also a root of this polynomial.

Therefore the polynomial in factored form is
`f(x)=(x-5)(x+3)(x-(-1+3i))(x-(-1+3i))`

We expand to get:
`f(x)=(x^2-2x-15)(x^2+2x+10)`

We expand further to get:\

`f(x)=x^4-9x^2-50x-150`