2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra:

If f(x) is a polynomial of degree n, where n > 0, f has a least one zero in the complex number system.

Linear Factorization Theorem:

If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors

Example1: Real Zeros of Polynomial Functions

• The first-degree polynomial f(x)= x-2 has exactly one zero: x=2
• Counting multiplicity, the second-degree polynomial function has exactly two zeros: x=3 and x=3

Example 2: Real and Complex Zeros of Polynomial Functions

• In some third degree polynomial functions there will be exactly three zeros, but the graph will only show the real zeros as intercepts.

Example 3: Finding the Zeros of a Polynomial Function

• To find the zeros, first find possible rational zeros then find the real zeros by graphing the equation, and finally use synthetic division to determine algebraically that the zeros are on the graph.

Complex Zeros Occur in Conjugate Pairs

Let f(x) be a polynomial function that has real coefficients. If a + bi, where b can't equal 0, is a zero of the function, the conjugate a-bi is also a zero of the function.

**this result is only true if the polynomial function has real coefficients.

Example 4: Finding a Polynomial with Given Zeros

• find a fourth-degree polynomial function with real coefficients, that has -1, -1, 3i as zeros
• from the Linear Factorization Theorem it can be written as
• or simply, let a=1 to get

Factors of a Polynomial:

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 with no real zeros is irreducible over the reals

for example:

**this quadratic is irreducible over the rationals, but reducible over the reals:

Example 5: Factoring a Polynomial

a. as the product of factors that are irreducible over the rationals

• first factor the polynomial into the product of two quadratic polynomials

b. as the product of linear factors and quadratic factors that are irreducible over the reals

• By factoring over the reals, you have:

Example 6: Finding the Zeros of a Polynomial Function

given that 1-3i is a zero of f

• because complex zeros occur in conjugate pairs, you know that 1-3i is also a zero of f.
• both x-(1+3i) and x-(1-3i) are factors of f(x). Multiplying these two factors produces [x-(1+3i)][x-(1-3i)]=[(x-1)-3i][(x-1)+3i]

• then you would use long division to divide into f(x) and then get
• therefore, you have
• and conclude that the zeros of f are 1+3i, 1-3i, 3 and -2