polynomial college algebra

Question

suppose f(x) is a degree 5 polynomial with rational coefficients that has three of its zeros as 2, -2i and 2+square root 5. write a possible f(x) polynomial. (sorry I didn't have the square root symbol on my keyboard) 

Answer 1

The 5 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.  You can pick A = 1, since you ask for one possible polynomial.  If the value of f(x) had been given at some point x₀, you would have to plug x₀ into f(x), set it equal to the given value, and then solve for A.

Answer 2

Two of these roots have conjugates:

 

x = 2i

x = 2 + √5

 

We can use the sum and product of the roots to find a quadratic factor:

x² - (sum of roots)x + product of roots

 

For the root, x=2i , we have the factor  (x² + 4).

 

For the root, x=2+√5, we have    x² - [(2 + √5) + (2 - √5)]x + (4 - 5) = (x² - 4x + 1) .

 

Then the last factor is (x - 2).  Put these factors all together to get the function

 

f(x) = (x - 2)(x² + 4)(x² - 4x + 1)

f(x) = (x³ - 2x² + 4x - 8)(x² - 4x + 1)