9. find a polynomial of degree 4 whose coefficients are real numbers and that has
zeros -1, 2, and -2i.

p(x) = - x³ + 2x² - 4x - 8

Step-by-step explanation:

Note that complex roots occur in conjugate pairs.

Given x = - 2i is a root then x = 2i is also a root

The roots of the polynomial are x = - 1, x = 2, x = - 2i and x = 2i

Thus the factors of the polynomial are

(x + 1), (x - 2), (x - 2i) and (x + 2i)

The polynomial is the product of it's roots, thus

p(x) = (x + 1)(x - 2)(x - 2i)(x + 2i) ← expand in pairs

      = (x² - x - 2)(x² - 4i²) → note i² = - 1, so

      = (x² - x - 2)(x² + 4) ← distribute parenthesis

      = + 4x² - x³ - 4x - 2x² - 8 ← collect like terms

      = - x³ + 2x² - 4x - 8

x coordinate can be written by using the formula, x= x1 + 2/5(x2 - x1)

step-by-step explanation:

formula

x = x1 + p/w(x2 - x1)

let the line segment ab with a(x1,y1) and b(x2,y2)

a point(x,y) divide ab in the ratio, 2: 3 the x coordinate can be written as

x = x1 + p/w(x2 - x1)

here p=2 and w = 2+3

therefore,

x coordinate can be written by using the formula, x= x1 + 2/5(x2 - x1)

similarly y coordinate y = y1 + 2/5(y2 - y1)