5.6 - The Fundamental Theorem of Algebra

5.6 - The Fundamental Theorem of Algebra

Objective

To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions

Recall

Last time we worked on factoring polynomial equations to find their solutions. You may have wondered, how did we know when to stop factoring? Like, is there a way to know exactly how many factors a polynomial has, so we can know if we have identified all of them?

Recall

In Algebra I, we learned three cases for the number of real-number solutions to a quadratic equation:

Recall

But in Algebra II, when we factored the quadratics we saw that each related equation has two roots:

We already know that a first-degree (linear) polynomial has one zero, the x-intercept. This suggests that the number of roots of a polynomial is related to its degree.

The Fundamental Theorem of Algebra

Multiple Choice

How many roots does the following function have?
f(x)= x⁵ - 3x³ + x

• 8
• 9
• 5
• 3

Multiple Choice

What is the least degree of the graph of the above polynomial?

• 3
• 4
• 5
• Not enough information 

Multiple Choice

A Polynomial of Degree 6 CAN have the following number of imaginary roots:

• 1
• 4
• 3
• 8

Poll

Ready to try some on your own?