2.5 the fundamental theorem of algebra students will use the fundamental theorem of algebra to...
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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.
Where are zeros that are complex numbers.
f x a x z x z x zn n( ) ( )( )...( ) 1 2
z z z1 2 3, ,
Example 1: Real Zeros of a Polynomial Function
Counting multiplicity, justify that the second-degree polynomial funcionhas exactly two factors and zeros.f x x x( ) 2 6 9
Example 2: Real and Imaginary Complex Zeros of a Polynomial Function
Justify that the third degree polynomial functionhas exactly three factors and zeros.
f x x x( ) 3 4
Example 3: Finding the zeros of a Polynomial Function
Write as the product of linear factors, and list all the zeros of f.
f x x x x x( ) 5 3 22 12 8y
x–2
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:
a) as a product of quadratic factors.
b) as a product of linear factors.
c) in complete factored form.
f x x x( ) 4 2 20
Example 5:Write as a product of quadratic factors, linear factors, in complete
factored form
y
x–2
2
f x x x( ) 4 2 20
Example 6: Finding the zeros of a polynomial function
Find all the zeros of given that 1 + 3i is a zero of f.
f x x x x x( ) 4 3 23 6 2 60
y
x–2
2
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