Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form?
7, -11, and 2 + 8i


f(x) = x4 - 9x3 - 56x2 + 290x - 5236

f(x) = x4 - 9x3 + 56x2 - 290x + 5236

f(x) = x4 - 145x2 + 580x - 5236

f(x) = x4 - 25x2 + 580x - 5236

Answer 1

We are given zeros of the polynomial : 7, -11, and 2 + 8i.

Note: The radical zero always comes with the pair of plus and minus sign.

Therefore, another zero would be 2-8i.

Now, in order to find the polynomial with the zeros 7, -11, 2 + 8i and 2-8i, we need to find the factors of the polynomial.

The factors of the polynomial would be (x-7)(x+11)(x-2-8i)(x-2+8i).

Let us multiply those factors to get the standard form of the polynomial.

$\left(x-2-8i\right)\left(x-2+8i\right)=\left(x-2-8i\right)\left(x-2+8i\right) =x^2-4x+68$

$\left(x-7\right)\left(x+11\right)=x^2+11x-7x-77$=$x^2+4x-77$

$\left(x^2-4x+68\right)\left(x^2+4x-77\right)=x^4-9x^2-16x^2+308x+272x-5236$

$=x^4-25x^2+580x-5236$.

Therefore, correct option is 4th option $f(x) = x^4 - 25x^2 + 580x - 5236$.