7, 7, 1 + i what is the polynomial function with real coefficients that has the given zeros

Question

the zeros are 7, 7, 1 + i

Answer 1

There has to be another complex zero to balance the given one. This zero is 1-i. The factors of the polynomial are (x-7)²(x-1+i)(x-1-i)=(x²-14x+49)(x²-2x+1+1)=

(x²-14x+49)(x²-2x+2)=

x⁴-2x³+2x²

  -14x³+28x²-28x

           +49x²-98x+98=

x⁴-16x³+79x²-126x+98. This is the lowest degree polynomial with the given factors. Other polynomials of the same degree can be found by multiplying this one by a constant. Higher degree polynomials can be found by multiplying by another factor of the type x+c where c is a constant.