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Section 3-5Finding Real Roots of Polynomial Equations
Objectives Identify the multiplicity of roots
Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations
As with some quadratic equations, factoring a polynomial equation is one way to find its real roots
Factoring Polynomials in Quadratic Form (Degree 4 or Higher)◦ Follow previous factoring rules◦ Remember rules of exponents when creating
binomials 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
Finding Real Roots of Polynomial Equations
4x6 + 4x5 – 24x4 = 0
x3 – 2x2 – 25x = –50
Solve the Polynomial Equation by Factoring
Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root.
In the root, 0, is a multiple root. ◦ For example, the root of 0 is a factor FOUR times
because . 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
Finding Real Roots of Polynomial Equations
You can use a graphing calculator to see roots You cannot always determine the multiplicity
of a roots from a graph. It is easiest to determine multiplicity when
the polynomial is in factored form. If you can’t factor, use synthetic division to
help you. Get the root for synthetic from the graphing
calculator
Finding the Real Roots of Polynomial Equations
x3 + 6x2 + 12x + 8 = 0
x4 + 8x3 + 18x2 – 27 = 0
x4 – 8x3 + 24x2 – 32x + 16 = 0
Identify the roots of each equation. State the multiplicity of each root
Not all polynomials are factorable, but the Rational Root Theorem can help you find all possible rational roots of a polynomial equation.◦ Also known as the “p over q” theorem
Finding Real Roots of Polynomial Equations
Use the Rational Root Theorem to list all possible rational roots.
Rational Root Theorem
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?
Marketing Application
Polynomial equations may also have irrational roots
Finding real Roots of Polynomial Equations
The Irrational Root Theorem says that irrational roots come in conjugate pairs◦ For example, if you know that is a root of x3 – x2
– 3x – 1 = 0, then you know that is 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
Finding Real Roots of Polynomial Equations
2x3 – 9x2 + 2 = 0
2x3 – 3x2 –10x – 4 = 0
Identify all of the Real Roots of a Polynomial Equation
DAY 1: p186 #15-26 Day 2: p187 #29-35, 40-44
Page186-188 #15-22, 24-35, 40-43
Accelerated: ADD #44-47
HOMEWORK