PLEASE HELP IN THE NEXT FEW MINUTES!!!

Question

Write a polynomial function f of least degree that has rational; coefficients a leading coefficient of 1 and zeros -3 and 5+i

Answer 1

There are 3 zeroes since complex solutions come in conjugate pairs.

x = -3 → x + 3 = 0

x = 5 + i → x - (5 + i) = 0

x = 5 - i → x - (5 - i) = 0

Let's distribute the complex roots.

[x - (5 + i)]*[x - (5 - i)] = x² - x(5 - i) - x(5 + i) + (5 + i)(5 - i)

= x² - 5x - ix - 5x + ix + 25 - 5i + 5i - i²

= x² - 10x + 26

What I did is to combine like terms. In other words, combine all the x's, i's, and constants. The i² is -1. So add 1 to 25 from there.

Distribute x + 3 to each term and combine like terms.

(x + 3)(x² - 10x + 26) = x³ - 10x² + 26x + 3x² - 30x + 78 = x³ - 7x² - 4x + 78