For the polynomial given below completely factor it over the

For the polynomial given below completely factor it over the real numbers. (If the expression is not factorable, enter NF.) x^2 - 8x + 41

Solution

We have to factorize the polynomial x² - 8x + 41 . We know that the roots of the polynomial ax² +bx +c are

[ - b ± ( b² – 4ac)] . Here, a = 1, b = -8 and c = 41. Therefore, the roots of the given polynomial are

[ -(-8) ± { ( -8)² - 4(1)(41) } ]/ 2*1 or,[ 8 ± ( 64 - 164)] / 2 or (8± - 100) / 2 or ( 8± 10i )/2 or 4± 5i . Then the given polynomial is ( x - 8 - 10i)( x - 8 + 10i). The zeros are 8+ 10i , 8 - 10i