Transcript roots - Coweta County Schools

Finding Real Roots of
Polynomial Equations
• How do we identify the multiplicity of
roots?
• How do we use the Rational Root
Theorem and the irrational Root Theorem
to solve polynomial equations?
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
In Lesson 3-5, you used several methods for
factoring polynomials. As with some quadratic
equations, factoring a polynomial equation is one
way to find its real roots.
Using the Zero Product Property, you can find the
roots, or solutions, of the polynomial equation
P(x) = 0 by setting each factor equal to 0 and
solving for x.
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
Example 1: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
4x6 + 4x5 – 24x4 = 0
Check using a graph.
2
4
4x  x  x  6  0
4 x  x  3  x  2   0
4
4x  0 x  3  0 x  2  0
4
x0
x  3
x2
The roots appear to be located at x = 0, x = –3, and x = 2. 
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
Example 2: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
x4 + 25 = 26x2
x  26 x  25  0
4
2
 x 2  25  x  1   0
 x  5   x  5  x  1   x  1   0
2
x  5  0 x  5  0 x 1  0 x 1  0
x  5
x5
x  1
The roots are 5, –5, 1, and –1.
Holt McDougal Algebra 2
x 1
Finding Real Roots of
Polynomial Equations
Example 3: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
2x6 – 10x5 – 12x4 = 0
2 x  x  5 x  6  0
4
2 x  x  6  x  1   0
4
2
2x  0 x  6  0 x 1  0
4
x0
x6
x  1
The roots are 0, 6, and –1.
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
Example 4: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
x3 – 2x2 – 25x = –50
3
2
x  2 x  25 x  50  0
x  x  2   25  x  2  0
2
x
2

 25  x  2   0
 x  5  x  5  x  2   0
x5  0 x5  0 x2  0
x5
x  5
Holt McDougal Algebra 2
x2
The roots are 5, –5, and 2.
Finding Real Roots of
Polynomial Equations
Sometimes a polynomial equation has a factor that
appears more than once. This creates a multiple root. In
3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3.
For example, the root 0 is a factor three times because
3x3 = 0.
The multiplicity of root r is the number of times that x – r
is a factor of P(x). When a real root has even multiplicity,
the graph of y = P(x) touches the x-axis but does not cross
it. When a real root has odd multiplicity greater than 1, the
graph “bends” as it crosses the x-axis.
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
You cannot always determine the multiplicity of a
root from a graph. It is easiest to determine
multiplicity when the polynomial is in factored
form.
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
Example 5: Identifying Multiplicity
Identify the roots of each equation. State the
multiplicity of each root.
2x4  12x3 + 18x2 = 0
2x  x  6 x  9  0
2
2 x  x  3 x  3   0
2
2x  0 x  3  0 x  3  0
2
x0
2
x3
x3
The root 0 has multiplicity of 2,
the root 3 has multiplicity of 2.
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
Example 6: Identifying Multiplicity
Identify the roots of each equation. State the
multiplicity of each root.
x3 – x2 – x + 1 = 0
2
x  x 1  1  x 1   0
x  1 x  1  0
2
 x  1  x  1  x  1  0
x 1  0 x 1  0 x 1  0
x 1
x  1
x 1
The root 1 has multiplicity 2, the root 1 has multiplicity 1.
Holt McDougal Algebra 2
Finding Real Roots of
Polynomial Equations
Lesson 4.1 Practice A
Holt McDougal Algebra 2