suppose f(x) is a degree 5 polynomial with rational coefficients that has three of it's zeros as 2, -2i and 2+5 write a possible f(x) polynomial

Question

college algebra

Answer 1

OK, with the additional information given, the zeros are

 

2

-2i

2i   [For the coefficients to be real, the complex conjugate must also be a zero.]

2 + √5

2 - √5   [The square roots come in ± pairs.]

 

The polynomial is 

 

f(x) = A(x - 2)(x + 2i)(x - 2i)[x - (2 + √5)][x - (2 - √5)]

 

where A is any rational number ≠ 0.  You may expand f(x) to standard form if you wish.  All of the coefficients should come out real and rational.

Answer 2

Perhaps you meant 2 + 5i; otherwise the zero is 2+5=7

 

Then 2i and 2 - 5i are also zeros.

 

The polynomial is (x-2)(x+2i)(x-2i)( x - (2 + 5i)) ( x - (2 - 5i)) = 0

 

                         (x-2)( x^2 + 4)( x^2 - (2 - 5i)x - (2 + 5i)x + (2+5i)(2 - 5i)) = 0

 

                         (x-2)(x^2 + 4) ( x^2 - 4x + (4 + 25)) = 0

                         (x-2)(x^2 + 4)(x^2 - 4x + 29) = 0