Date post: | 19-Jan-2016 |
Category: |
Documents |
Upload: | mildred-warner |
View: | 224 times |
Download: | 1 times |
Holt McDougal Algebra 2
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
Solve the polynomial equation by factoring.
Example 1: Using Factoring to Solve Polynomial Equations
4x6 + 4x5 – 24x4 = 0 44x 2x 6x
x x3 2 0 4 4 x
0
04 4 x 03 x 02 x0x 2x3x
Check using a graph.
The roots appear to be located at x = 0, x = –3, and x = 2.
Holt McDougal Algebra 2
Finding Real Roots of Polynomial Equations
Solve the polynomial equation by factoring.
Example 2: Using Factoring to Solve Polynomial Equations
x4 + 25 = 26x2
02526 24 xx
0 2x2x 25 1
0 1x1x5x5x
01 x 01 x05 x05 x5x 5x 1x 1x
The roots are 5, –5, 1, and –1.
Holt McDougal Algebra 2
Finding Real Roots of Polynomial Equations
0
Solve the polynomial equation by factoring.
Example 3: Using Factoring to Solve Polynomial Equations
2x6 – 10x5 – 12x4 = 0 2x 642x x5
02 4 x 06 x 01 x0x 6x 1x
0 2 4 x x x 16
The roots are 0, 6, and –1.
Holt McDougal Algebra 2
Finding Real Roots of Polynomial Equations
02 x
0
Solve the polynomial equation by factoring.
Example 4: Using Factoring to Solve Polynomial Equations
x3 – 2x2 – 25x = –50
050252 23 xxx
The roots are 5, –5, and 2.
2 25 22x
252 x 02 x
5x5x02 x05 x05 x
2x5x5x
x x
Holt McDougal Algebra 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
0 2 2 x
Identify the roots of each equation. State the multiplicity of each root.
Example 5: Identifying Multiplicity
2x4 12x3 + 18x2 = 0
0 2x
0x
x x
The root 0 has multiplicity of 2, the root 3 has multiplicity of 2.
22x x6 933
03 x03 x02 2 x3x3x
Holt McDougal Algebra 2
Finding Real Roots of Polynomial Equations
x3 – x2 – x + 1 = 0
Identify the roots of each equation. State the multiplicity of each root.
Example 6: Identifying Multiplicity
0 2x 1 1 1 01 x 12 x
1x1x01 x01 x01 x
1x1x1x
01 x
The root 1 has multiplicity 2, the root 1 has multiplicity 1.
x x
Holt McDougal Algebra 2
Finding Real Roots of Polynomial Equations
Lesson 4.1 Practice A