Download 1.3 Graphs of Functions - East Peoria Community High School

2.5 The Fundamental Theorem of Algebra
Students will use the fundamental theorem of algebra
to determine the number of zeros of a polynomial.
Students will find all zeros of polynomial functions,
including complex zeros.
Students will find conjugate pairs of complex zeros.
Students will find zeros of polynomials by factoring.
The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, then f has
at least one zero in the complex number
system.
* Remember that the polynomial has at
most n zeros.
Linear Factorization Theorem
If f(x) is a polynomial of degree n, f has
precisely n linear factors.
f ( x )  an ( x  z1 )( x  z2 )...( x  zn )
Where z1 , z2 , z3 are zeros that are complex
numbers.
Example 1: Real Zeros of a Polynomial Function
Counting multiplicity, justify that the second-degree polynomial funcion
f ( x)  x 2  6x  9 has exactly two factors and zeros.
Example 2: Real and Imaginary Complex Zeros of a
Polynomial Function
Justify that the third degree polynomial function
has exactly three factors and zeros.
f ( x)  x 3  4 x
Example 3: Finding the zeros of a Polynomial
Function
Write f ( x)  x5  x 3  2 x 2  12 x  8 as the product of linear factors, and
list all the zeros of f.
y
2
x
–2
Example 4: Finding a Polynomial with Given Zeros
Find a fourth degree polynomial function with real
coefficients that has - 1, - 1, and 3i as zeros.
Example 5: Factoring a Polynomial
Write the polynomial: f ( x)  x 4  x 2  20
a)
as a product of quadratic factors.
b)
as a product of linear factors.
c)
in complete factored form.
Example 5:
f ( x)  x 4  x 2  20
Write as a product of quadratic factors, linear factors, in complete
factored form
y
2
x
–2
Example 6: Finding the zeros of a polynomial
function
Find all the zeros of
f ( x)  x 4  3x 3  6x 2  2 x  60
1 + 3i is a zero of f.
y
given that
2
x
–2