Question

1. write a polynomial function of minimum degree with real coefficients whose zeros are -1 and 2 ± 5i. write the polynomial in standard form.

a) $f(x) = x^3 - 3x^2 + 25x + 29$ b) $f(x) = x^3 - 3x^2 + 25x - 29$
c) $f(x) = x^3 - 3x^2 + 25x + 30$ d) $f(x) = x^3 + 3x^2 + 25x + 29$

Explanation:

Step1: Recall Complex Conjugate Root Theorem

If a polynomial has real coefficients and a complex root \(a + bi\), then its conjugate \(a - bi\) is also a root. So the roots are \(x=-1\), \(x = 2 + 5i\), and \(x=2 - 5i\).

Step2: Form the Polynomial from Roots

The polynomial can be written as \(f(x)=(x + 1)[x-(2 + 5i)][x-(2 - 5i)]\). First, simplify the quadratic factor:

$$
LATEXBLOCK0
$$

Step3: Multiply by the Linear Factor

Now multiply by \((x + 1)\):

$$
LATEXBLOCK1
$$

Answer:

a) \(f(x)=x^3 - 3x^2 + 25x + 29\)