find a ploynomial fucnction of least degree having only real

find a ploynomial fucnction of least degree having only real coefficients, a leading coefficient of 1, and zeros of 5 and 4+i

Solution

coefficients are real , as 4+i is root, then its conjugate 4-i is also the root

zero are 5 and 4+i ,4-i

ploynomial function of least degree P(x) =(x-5)(x-(4+i))(x-(4-i))

P(x) =(x-5)((x-4)-i)((x-4)+i))

P(x) =(x-5)((x-4)²-i²)

P(x) =(x-5)((x²-8x+16)+1)

P(x) =(x-5)(x²-8x+17)

P(x) =x(x²-8x+17) -5(x²-8x+17)

P(x) =x³-8x²+17x -5x²+40x-85

P(x) =x³-13x²+57x -85