Transcript Lesson 2.5 The Fundamental Theorem of Algebra

Lesson 2.5
The Fundamental Theorem
of Algebra
For f(x) where n > 0, there is at least one zero in the complex
number system
Complex → real and imaginary
Real zeros → can see on graph
Imaginary → cannot see on graph
f(x) of degree n will have exactly n zeros (real and imaginary)
Complex zeros always occur in conjugate pairs
“If a + bi is a zero, then a – bi is also”
How do we find the zeros of
f ( x)  x  x  2 x  12 x  8
5
3
2
For a polynomial f(x) with degree n:
1) Find real zero(s) using calculator or Factor Theorem and
Synthetic Division
 If there are n real roots, stop
2) Divide out real zeros to obtain a 2nd degree equation
3) Solve 2nd degree equation for x (use Quadratic Equation if
necessary)
Example 1
f ( x)  x  x  2 x  12 x  8
5
3
2
Example 2
f ( x)  x  6 x  10 x  6 x  9
4
3
2
Problem Set 2.5