Find an polynomial of degree four, with roots given x =1, x = 4, x = -4+2i, x = -4-2i, and passes through (3, 318)

Answer 1

First, from the given x values, find the 4 factors:

x-4, x-1, x+4+2i, x+4-2i

Multiply the 4 factors---one way is to do it in pairs:

(x-4) * (x-1) = x² - 5x + 4

(x+4+2i) * (x+4-2i) = x² + 8x + 20

Then multiply the 2 quadratics to get:

y = x⁴ + 3x³ - 16x² - 68x + 80

Finally, plug in x=3, and get y = -106. We want y to be +318, so multiply the entire expression by - 3 to get:

y = -3x⁴ - 9x³ + 48x² + 204x - 240

CHECK THIS---plug in x=3 and make sure you get y=318