FUNDAMENTAL THEOREM OF ALGEBRA AND FINDING REAL ROOTSHonors Advanced AlgebraPresentation 2-6

WARM-UP Given the roots, write the factors of the

quadratic and the polynomial.1. x = -3, 2

2. x = 4, 3

3. x = -5, 5

4. x = 1, 1

ROOTS OF A POLYNOMIALThe places where a polynomial

crosses or touches the x-axis are called the roots of the polynomial.

They are also known as x-intercepts, zeros, or solutions.

Roots can be found by setting the polynomial equal to 0 and solving.

THE FUNDAMENTAL THEOREM OF ALGEBRA Every polynomial function of degree has at

least one zero, where a zero may be a complex number.

Corollary: Every polynomials function of degree has exactly n zeros, including multiplicities and irrational roots.

MULTIPLICITY OF A ROOT A root can occur once or multiple times.

If the root repeats, the number of times is known as the multiplicity of a root.

If the multiplicity is even, the graph will touch the x-axis but not cross it; if the multiplicity is odd, the graph will intersect the x-axis.

Example: Write a polynomial with roots 1 with a multiplicity of 2 and -3 with a multiplicity of 1.

MULTIPLICITY OF A ROOT Multiplicity of 1

Multiplicity of 2

Multiplicity of 3

MULTIPLICITY OF A ROOT Example 2: Write a possible polynomial as

the product of factors given the graph below.

RATIONAL ROOTS THEOREM If the polynomial P(x) has integer

coefficients, then every rational root of the polynomial equation P(x) = 0 can be written in the form , where p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).

RATIONAL ROOTS THEOREM Example: Find all possible rational roots of

RATIONAL ROOTS THEOREM Example: Use the rational roots theorem and

a graph to completely factor

IRRATIONAL ROOTS THEOREM If the polynomial P(x) has rational

coefficients and is a root of the polynomial equation P(x) = 0, where a and b are rational and is irrational, then is also a root of P(x) = 0.

IRRATIONAL ROOTS THEOREM Irrational roots always come in

conjugate pairs.

Example: If 5 is a root of a polynomial, what is another root of the polynomial? What is the corresponding factor?

COMPLEX CONJUGATE ROOT THEOREM If a + bi is a root of a polynomial equation

with real-number coefficients, then a – bi is also a root.

Imaginary roots always come in conjugate pairs.

Example: If 3 + 2i is a root of a polynomial, what is another root of the polynomial? What is the corresponding factor?

EXAMPLE OF FINDING ROOTS Solve by finding all roots. Step 1: Use the Rational Roots Theorem to

identify possible rational roots.

EXAMPLE OF FINDING ROOTS Solve by finding all roots. Step 2: Graph the polynomial to narrow down

your options.

EXAMPLE OF FINDING ROOTS Solve by finding all roots. Step 3: Test the possible real roots to help

factor the polynomial down to a quadratic. Then find remaining zeros.

EXAMPLE OF WRITING A POLYNOMIAL FUNCTION GIVEN ZEROS Write the simplest polynomial function with

zeros 1 + i, , and -3.

HOMEWORKP. 121, #24-25, 30-35Pg. 127-128, #12-32 even, 38-43