Find a polynomial of minimum degreee with a leading coeffici

Find a polynomial of minimum degreee with a leading coefficient of 1 that has the given zeros

2, 5-i, 5+i

Solution

The polynomial of minimum degree with a leading coefficient of 1and having 2, 5 -i and 5 +i as zeros is ( x-2)[x- (5-i)][x -(5+i)] = (x-2)[ (x-5)+ i ][(x-5) - i] = (x-2)[ (x-5)² -i² } ( as (a+b)(a-b) = a² - b² ; here a = x-5 and b = i)

Or, (x-2)( x²-10x+25 +1) or, (x-2)(x²-10x+26) or, x³ -10x² +26x - 2x² +20x -52 or , x³ -12x²+46x -52 ( as i² = -1)