Construct a 5th degree polynomial having real coefficients w

Construct a 5^th degree polynomial having real coefficients with zeros 2i, -5i, and 1. (Express your answer in the form P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f.)

Solution

solution

To get the polynomial you are looking for there are a few pieces of information that we need to add.

ALL COMPLEX ROOTS COME IN PAIRS. So they have purposefully left out two roots to see if you know that you must find the \"other\" root for each complex number.

The two roots are complex conjugates. That means that if one is \"2i\", the other is \"-2i\". If you is \"1\", the other is \"-1\" and -5i

Now we have 5 roots and can make a fifth degree polynomial.

The roots are:

x=2i, x= -2i, x=-5i, x=1, x= -1

For each \"x=\", move everything to the left side so that these become factors instead of zeros.

x-2i=0

x+2i=0

x+5i=0

x-1=0

x+1=0

Now your factors are:

(x-2i)(x+2i)(x+5i)(x-1)(x+1)

Multiply the first two and get

x^2-4i^2= x^2+4

The 4th and 5th terms multiplied together give:

(x-1)(x+1)

x^2-1

Now you have all real coefficients to work with:

Your original :(x-2i)(x+2i)(x+5i)(x-1)(x+1)

Has become:(x^2+4)(x^2-1)(x+5i)

Multiply the first two. Then use that result to multiply with the remaining factor (x+5i)

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

(x^4+3x^2-4)(x+5i)

x^5+5ix^4+3x^3+15ix^2-4x-20i

x^5+5ix^4+3x^3+15ix^2-4x+20i