Download 3-5 finding real zeros

CHAPTER 3
3-5 Finding the real roots of polynomial
Equations
SAT Problem of the day
• If f(3)=9 ,then 𝑓 −1 4 =
• A)-2
• B)0
• C)2
• D)16
• E)can not be determined
Solution to the SAT Problem
• Right Answer: E
objectives
Identify the multiplicity of roots.
Use the Rational Root Theorem and the irrational
Root Theorem to solve polynomial equations.
Real Zeros
• In previous lessons, you used several methods for
factoring polynomials. As 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.
Example#1
• Solve the polynomial equation by factoring
• 4x6 + 4x5 – 24x4 = 0
Example#2
• Solve the polynomial equation by factoring.
• x4 + 25 = 26x2
Example#3
• Solve the polynomial equation by factoring.
• 2x6 – 10x5 – 12x4 = 0
Student guided practice
• Do problems 2-5 in your book page 186
Real Zeros
• 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.
Multiplicity
• What is multiplicity?
• Answer: 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.
Example
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.
Example#4
• Identify the roots of each equation. State the
•
•
•
•
multiplicity of each root.
x3 + 6x2 + 12x + 8 = 0
Solution:
x3 + 6x2 + 12x + 8 = (x + 2)(x + 2)(x + 2)
x + 2 is a factor three times. The root –2 has a multiplicity
of 3.
Example#5
• Identify the roots of each equation. State the
•
•
•
•
multiplicity of each root.
x4 + 8x3 + 18x2 – 27 = 0
Solution:
x4 + 8x3 + 18x2 – 27 = (x – 1)(x + 3)(x + 3)(x + 3)
x – 1 is a factor once, and x + 3 is a factor three times.
The root 1 has a multiplicity of 1. The root –3 has a
multiplicity of 3.
Student guided practice
• Do problems 8 and 9 in your book page 186
Rational Theorem
• Not all polynomials are factorable, but the Rational Root
Theorem can help you find all possible rational roots of a
polynomial equation.
• What is the rational zero?
Application
• The design of a box specifies that its length is 4
inches greater than its width. The height is 1 inch less
than the width. The volume of the box is 12 cubic
inches. What is the width of the box?
Application
• A shipping crate must hold 12 cubic feet. The width
should be 1 foot less than the length, and the height
should be 4 feet greater than the length. What should
the length of the crate be?
Irrational root theorem
• Polynomial equations may also have irrational roots.
Irrational root theorem
• The Irrational Root Theorem say that irrational roots come
in conjugate pairs. For example, if you know that 1 +
a root of x3 – x2 – 3x – 1 = 0, then you know that 1 –
also a root.
Recall that the real numbers are made up
of the rational and irrational numbers. You
can use the Rational Root Theorem and
the Irrational Root Theorem together to find
all of the real roots of P(x) = 0.
is
is
Example
• Identify all the real roots of 2x3 – 9x2 + 2 = 0.
Example
• Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0.
Student guided practice
• Do problems 11-14 in your book page 186
Homework!!
• Do problems 15-18,21,24,29-31
Closure
• Today we learned how to fins real zeros
• Next class we are going to learn about the fundamental
theorem of algebra.