Complex Conjugate Root (Zero) Theorem

Conjugate Root (Zero) Theorem - How it Works - Video

Conjugate Root (Zero) Theorem

Conjugate Root (Zero) Theorem:

Here we have the conjugate root (zero) theorem. This theorem tells us the complex numbers work in pairs when dealing with zeros. If there is 1 complex number, then you know there has to be at least 2 complex numbers. If there is 3 complex numbers, then there has to be at least 4 complex numbers.

The line above indicates that is the conjugate so z = a + bi has a conjugate of z ̅= a - bi, and z = a - bi has a conjugate of  z ̅= a + bi.

What are Square Roots?

What are Square Roots?

Before we continue talking about complex numbers. We need to talk about what are square roots? The definition is a value that when multiplied by itself gives the number. In the example, 2 * 2 equals 4 but now we know that -2 * -2 also equals 4. That is why there is ± symbol in front of a square root when we solve an equation.

Quadratic Formula

Quadratic Formula:

Now that we had that quick refresher on square roots. Let's bring out the heavy artillery, the quadratic formula. You see that ± in the numerator. There is where it comes from. We have to two ways when multiplying to get a result. We can multiply a positive times a positive or a negative times a negative. 

That is complex numbers have a pair (conjugate). One will have a positive sign in front of the i and one will have a negative sign in front the i.

Example 1

Example 1:

We have 1 zero and we need to express our function as a product of linear and quadratic polynomials. We also know that it is degree 2. So we know that there are 2 zeros in total.

Since we have a complex number, we know that is has a conjugate. We have the given z1 = -3i, which means z ̅2 = 3i. Now let's put the zeros in parentheses and simplify.

So our answer is x2 + 9.

Example 2

Example 2:

We have 1 zero and we need to express our function as a product of linear and quadratic polynomials. We also know that it is degree 2. So we know that there are 2 zeros in total.

Since we have a complex number, we know that is has a conjugate. We have the given z1 = -1 + 4i, which means z ̅2 = -1 - 4i. Now let's put the zeros in parentheses and simplify.

So our answer is x2 + 2x + 17.

Example 3

Example 3:

We have 2 zeros and we need to express our function as a product of linear and quadratic polynomials. We also know that it is degree 3. So we know that there are 3 zeros in total.

Since we have a complex number, we know that is has a conjugate. We have the given z1 = -4, z2 = 5i, which means z ̅3 = -5i. Now let's put the zeros in parentheses and simplify.

So our answer is (x + 4) * (x2 + 25).

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