Write a polynomial of least degree with rational coefficients so that P(x)=0 has the given root. 5-7i

Answer 1

If 5 - 7i is root, then so is its conjugate 5 + 7i.

If those complex numbers are roots of the polynomial function, then the following are factors:

[x - (5 - 7i)] [x - {5 + 7i)]

The product of those factors can be rewritten as:

[ (x - 5) + 7i][(x - 5) - 7i]

Since that is the product of the sum and difference of the same two numbers (x - 5) and 7i, the result is a difference of squares.

(x - 5)² - (7i)²

x² - 10x + 25 - 49i²

Replacing i² with -1 results in:

x² -10x +25 +49

So, P(x) = x² - 10x + 74

You can check the result with either the quadratic formula or by completing the square.

x² -10x + 74 = 0

x² -10x = -74

x² - 10x +25 = -74 +25 (complete the square)

(x - 5)² = -49

x - 5 =±7i (square root property)

x = 5 ± 7i