2.5- Fundamental Theorem of Algebra

2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

Definition- The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Including polynomials with real coefficients, since every real number is a complex number with no imaginary part.

-Through the use of the Fundamental theorem of Algebra you obtain the Linear Factorization Theorem

Linear factorization Theorem

Definition- If f (x) is a polynomial of a degree “n”, where n is greater than 0, “ f ” has at least one zero in the complex number system.

-          If f(x) is a polynomial of degree “n” where “n is greater than zero”, “f” has precisely “n” linear factors.

-          f (x) = an (x – c^1)( x – c^2)( x – c^3) … ( x – c^n)

-          wherec1, c2, c3, … cn are complex numbers

-This Theorem does not tell one how to find the zeroes or factors of a polynomial.

-EX.

- f (x) = t4 - 5t3 + 15t2 - 45t + 54

Step 1: find all the possible zeroes of the function:

(factors of 54/factors of 1) = ±1, ± 2, ± 3, ±6, ± 9, ± 18, ± 27, ± 54

Step 2: Use synthetic division to find out which ones are factors

.

Step 3: Use synthetic division to find another factor.

f (x) = (x – 2)(x – 3)(x^2 + 9)

0 = (x – 2)(x – 3)(x^2 + 9)

Step 4: Write this function in linear form:

f(x) = (x – 2)(x – 3)(x + 3i)(x – 3i)

Step 5: Find the zeros of “f”

x = {2, 3, 3i, and -3i}

Complex Zeros Occur in Conjugate Pairs

Definition- Let f(x) be a polynomial function that has real coefficents. If a+ bi, Where b is not equal to 0, is a zero of the function, the conjucate a – bi is also a zero of the function

            -Be sure that the result is true only if the polynomail function has real coefficients

-For instance the result applies to the function f(x)=x^2 + 1, but not to the function g(x)=x- i

-EX.

-          2 – 3i is a zero of p(x) = x^3 – 3x^2 + 9x + 13 as shown here:

p(2 – 3i) = (2 – 3i)^3 – 3(2 – 3i)^2 + 9(2 – 3i) + 13

             = (–46 – 9i) – 3(–5 – 12i) + (18 – 27i) + 13

            = –46 – 9i + 15 + 36i + 18 – 27i + 13

             = 0.

-          By the conjugate pair theorem, 2 + 3i is also a zero of p(x).

p(2 + 3i) = (2 + 3i)^3 – 3(2 + 3i)^2 + 9(2 + 3i) + 13

             = (–46 + 9i) – 3(–5 + 12i) + (18 + 27i) + 13

             = –46 + 9i + 15 – 36i + 18 + 27i + 13

             = 0.

Factoring a Polynomial

-The Linear Factorization Theorem shows that you can write any nth degree polynomial as the product of n linear factors.

-Every Polynomial of degree n> 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros.

            -A quadratic factor with no real zeros is said to be irreducible over the reals.

-EX.

            -2x² + 4x  

= 2x(x + 2)