Download 2-6 – Fundamental Theorem of Algebra and Finding Real Roots

FUNDAMENTAL THEOREM OF
ALGEBRA AND FINDING REAL
ROOTS
Honors Advanced Algebra
Presentation 2-6
WARM-UP

1.
Given the roots, write the factors of the quadratic
and the polynomial.
x = -3, 2
2.
x = 4, 3
3.
x = -5, 5
4.
x = 1, 1
ROOTS OF A POLYNOMIAL
 The
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 𝑛 ≥ 1 has at
least one zero, where a zero may be a complex
number.
Corollary: Every polynomials function of degree
𝑛 ≥ 1 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
𝑝
written in the form , where p is a
𝑞
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
𝑃 𝑥 = 4𝑥 4 − 21𝑥 3 + 18𝑥 2 + 19𝑥 − 6
RATIONAL ROOTS THEOREM

Example: Use the rational roots theorem and a
graph to completely factor
𝑃 𝑥 = 4𝑥 4 − 21𝑥 3 + 18𝑥 2 + 19𝑥 − 6
IRRATIONAL ROOTS THEOREM
 If
the polynomial P(x) has rational
coefficients and 𝑎 + 𝑏 𝑐 is a root of the
polynomial equation P(x) = 0, where a and
b are rational and 𝑐 is irrational, then
𝑎 − 𝑏 𝑐 is also a root of P(x) = 0.
IRRATIONAL ROOTS THEOREM
 Irrational
roots always come in conjugate
pairs.
 Example:
If 5 − 2 3 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 𝑥 4 + 𝑥 3 + 2𝑥 2 + 4𝑥 − 8 = 0 by finding all
roots.
 Step 1: Use the Rational Roots Theorem to
identify possible rational roots.

EXAMPLE OF FINDING ROOTS
Solve 𝑥 4 + 𝑥 3 + 2𝑥 2 + 4𝑥 − 8 = 0 by finding all
roots.
 Step 2: Graph the polynomial to narrow down
your options.

EXAMPLE OF FINDING ROOTS
Solve 𝑥 4 + 𝑥 3 + 2𝑥 2 + 4𝑥 − 8 = 0 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, 2, and -3.
HOMEWORK
P.
121, #24-25, 30-35
Pg. 127-128, #12-32 even, 38-43