Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. (2 points)
3, -13, and 5 + 4i

f(x) = x4 - 8x3 - 12x2 + 400x - 1599

f(x) = x4 - 200x2 + 800x - 1599

f(x) = x4 - 98x2 + 800x - 1599

f(x) = x4 - 8x3 + 12x2 - 400x + 1599

Answer 1

p(x)=$x^{4}$ - $98x^{2}$ + 800x - 1599

given x = a is a zero of a polynomial p (x) then

(x - a) is a factor of the polynomial

complex zeros occur in conjugate pairs

5 + 4i is a zero hence 5 - 4i is a zero

factors are (x - 3),(x + 13),(x - (5 + 4i) and (x - (5 - 4i)

The polynomial is the product of it's factors

p(x) = (x - 3)(x + 13)(x - (5 + 4i)(x - (5 - 4i)

expanding the factors

p(x) = ($x^{2}$ + 10x - 39)($x^{2}$ - 10x + 41)

    = $x^{4}$ - $98x^{2}$ + 800x - 1599