The Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n, then P(x) = 0 has exactly n roots, including multiple and complex roots.
What does this mean?
Important note: Irrational and imaginary zeros come in ___________________.
Class Example:
Find all zeros of the function y = x4 - 5x3 + 10x2 – 20x + 24
Step One: Graph and investigate the function.
1.) How many zeros will this function have? 2.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Station 1:
Find all zeros of the function 𝑓 𝑥 =
Step One: Graph and investigate the function.
3.) How many zeros will this function have? 4.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Station 2:
Find all zeros of the function 𝑓 𝑥 =
Step One: Graph and investigate the function.
1.) How many zeros will this function have? 2.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Station 3:
Find all zeros of the function 𝑓 𝑥 =
Step One: Graph and investigate the function.
1.) How many zeros will this function have? 2.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Station 4:
Find all zeros of the function 𝑓 𝑥 =
Step One: Graph and investigate the function.
1.) How many zeros will this function have? 2.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Station 5:
Find all zeros of the function 𝑓 𝑥 =
Step One: Graph and investigate the function.
3.) How many zeros will this function have? 4.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Station 6:
Find all zeros of the function 𝑓 𝑥 =
Step One: Graph and investigate the function.
3.) How many zeros will this function have? 4.) View the graph. How many real roots are there? How many imaginary roots are there? Step Two: Find all RATIONAL zeros. (zeros that can be written as an integer or fraction, NOT the irrational never-­‐ending mess!)
Step Three: Break down the function using rational zeros and synthetic division. (sometimes multiple divisions!)
Step Four: Use another method (quadratic formula, factoring, square roots) to solve the remaining
polynomial.
Step Five: Write out all the zeros and check that your original investigational work is confirmed.
Find all the zeros of the following functions.
1. 𝑓 𝑥 = 2𝑥 ! + 3𝑥 ! + 18𝑥 + 27
2. 𝑓 𝑥 = 𝑥 ! + 64
3. 𝑓 𝑥 = 𝑥 ! + 4𝑥 ! + 7𝑥 + 28
4. 𝑓 𝑥 = 2𝑥 ! + 𝑥 ! + 1