Form the fifthdegree polynomial function with real coefficie

Form the fifth-degree polynomial function with real coefficients sketched here given that 1 + i is a zero. F(X) = (Simplify your answer. Use integers or fractions for any numbers in the expression Type an expression use the variable.)

Solution

1+i is a zero

then 1-i would be otheer zero since complex zeros are in pairs

one zero is -2 with odd multiplicity

one zero is 1 with even multiplicity

therefore , polynomial would be

y = a(x-(1+i)) ( x - (1-i))(x-(-2))(x-1)^2

y = a ( x-1-i)(x-1+i)(x+2)(x-1)^2

y intercept = (0,3)

3 = a (-1-i)(-1+i)(2)(1)

3 = a (4)

a = 3/4

hence polynomial is

p(x) = 3/4( x^5-2x^4-x^3+8x^2-10x+4)