3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Bellwork 10-28-14
Factor completely.
1. 2y3 + 4y2 – 30y
Solve each equation.
2. x2 – 9 = 0
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Objectives
Identify the multiplicity of roots.
Use the Rational Root Theorem and the irrational Root
Theorem to solve polynomial equations.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Vocabulary
multiplicity
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
With some quadratic equations, factoring a polynomial
equation is one way to find its real roots.
Recall 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.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Example 1A: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
4x6 + 4x5 – 24x4 = 0
Factor out the GCF, 4x4.
Factor the quadratic.
Set each factor equal to 0.
Solve for x.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Example 1A Continued
Check Use a graphing
calculator. The roots appear
to be located at x = 0, x = –3,
and x = 2.
Y1=4x6 + 4x5 – 24x4
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Example 1B: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
x4 + 25 = 26x2
Set the equation equal to 0.
Factor the trinomial in quadratic form.
Factor the difference of two squares.
Solve for x.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
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 xaxis.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Not all polynomials are factorable, but the Rational Root
Theorem can help you find all possible rational roots of a
polynomial equation.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
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.
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Example 2A: Identifying Multiplicity
Identify the roots of each equation. State the multiplicity of
each root.
x3 + 6x2 + 12x + 8 = 0
Check Use a graphing calculator. A
calculator graph shows a _____
near (–2, 0).
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3­5 Finding Real Roots of Polynomial Equations­ period 1.notebook
October 28, 2014
Homework: Worksheet Practice A #'s 1-11
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