6-5 & 6-6
Finding All Roots of Polynomial Equations
Warm Up: Factor each expression completely.
1. 2y3 + 4y2 – 30
2. 3x4 – 6x2 – 24
LEARNING GOALS – LESSON 6.5 & 6.6 DAY 1
6.5.1: Use factoring to solve polynomial equations.
6.5.2: Use a graph and synthetic division to identify all real roots of a
polynomial equation.
6.6.1: Use a graph and synthetic division to find all roots (irrational and
imaginary) of a polynomial equation.
Factoring a polynomial equation is one way to find its real roots.
You can find the roots, or solutions, of the polynomial equation P(x) = _____ by
factoring P(x) and using the _________ ____________ ________________.
6-5 Example 1: Using Factoring to Solve Polynomial Equations
Solve the polynomial equation by factoring.
A. 4x6 + 4x5 – 24x4 = 0
Multi-Step Factoring Check List
1.
Put the equation in Standard Form = 0
2.
Factor out a GCF
3.
Count terms to choose a factoring method:
2 Terms: Difference of Squares
3 Terms: Easy Method or AC Method
4 Terms: Grouping
4.
Check each factored binomial for Difference
of Squares
5.
Use the Zero Product Property for roots
6-5 & 6-6
Finding All Roots of Polynomial Equations
6-5 Example 1: Using Factoring to Solve Polynomial Equations Contd.
Solve the polynomial equation by factoring.
B.
x4 + 25 = 26x2
C.
x3 – 2x2 – 25x = –50
6-5 Example 2: Identify All of the Real Roots of a Polynomial Equation
A. Identify all the real roots of the polynomial equation.
2x3 – 3x2 –10x – 4 = 0.
Step 1: Use the calculator to graph the
polynomial. Use the table function and graph to
find an integer x-intercept.
Step 2 Use synthetic division to divide
the integer you found in Step 1
into 2x3 – 3x2 –10x – 4.
Step 3 Solve the resulting equation,
_________________________= 0 to
find the remaining roots.
2
–3
–10 – 4
6-5 & 6-6
Finding All Roots of Polynomial Equations
6-5 Example 2: Identify All of the Real Roots of a Polynomial Equation
B. Identify all the real roots of the polynomial equation.
2x3 – 9x2 + 2 = 0.
Step 1: Use the calculator to graph the
polynomial. Use the table function and graph to
find an integer x-intercept.
Step 2: Use synthetic division to divide
the integer you found in Step 1
into _____________________.
Step 3 Solve the resulting equation,
_________________________= 0 to
find the remaining roots.
6-6 Example 3: Finding All Roots (Irrational and Imaginary) of a Polynomial
A. Solve the polynomial equation by finding all roots.
x4 – 3x3 + 5x2 – 27x – 36 = 0
Step 1: Use the calculator to graph the
polynomial. Use the table function and graph to
find an integer x-intercept.
Step 2: Use synthetic division to divide
the integer you found in Step 1
into _____________________.
Step 3 Solve the resulting equation,
_________________________= 0 to
find the remaining roots.
6-5 & 6-6
Finding All Roots of Polynomial Equations
6-6 Example 3: Finding All Roots (Irrational and Imaginary) of a Polynomial
A. Solve the polynomial equation by finding all roots.
x4 + 4x3 – x2 + 16x – 20 = 0
Step 1: Use the calculator to graph the
polynomial. Use the table function and graph to
find an integer x-intercept.
Step 2: Use synthetic division to divide
the integer you found in Step 1
into _____________________.
Step 3 Solve the resulting equation,
_________________________= 0 to
find the remaining roots.
6-5 & 6-6
Finding All Roots of Polynomial Equations
Warm-up:
Fill in each of the statements to make them true using the given information.
GIVEN:
1.)
If 5 is a root of P(x) = 0 then . . .
P(5) = ______
3.) (x - ___) is a factor of P(x)
2.) 5 si an ____- intercept of the graph of
P(x)
4.) When you divide P(x) by (x – ____)
you get a remainder of _____
5.) 5 is a ___________ of the graph of P(x)
LEARNING GOALS – LESSON 6.5 & 6.6 DAY 2
6.5.3: Define and identify multiplicity of roots of polynomial functions in
factored form or from a graph.
6.6.2: Write a polynomial function given all rational zeros.
6.6.3: Use understanding of conjugate irrational and imaginary root pairs to
write polynomials given some of their zeros.
Sometimes a polynomial equation has a factor that appears more than once.
This creates a ______________ root.
FACTOR:
FACTORED FORM:
ZERO PRODUCT PROPERTY:
ROOTS:
3x5 + 18x4 + 27x3 = 0
3x3 (x + 3) (x + 3) = 0 OR 3x3 (x + 3)2 = 0
3x3 = 0 (x + 3) = 0
THE FUNDAMENTAL THEOREM OF ALGEBRA
Every polynomial function of degree n of 1 or more has EXACTLY n zeros
including multiplicity.
Degree of Polynomial: ________
Degree of factor 3x3: _________
Degree of factor (x + 3): ______
Multiplicity of root x = _____;______
Multiplicity of root x = _____;______
6-5 & 6-6
Finding All Roots of Polynomial Equations
Identify the roots of each equation. State the multiplicity of each root.
A. 2x6 – 10x5 – 12x4 = 0
B. x 4 – 13x2 + 36 = 0
MOST OF THE TIME (like in your homework) the polynomials with only integer
roots are NOT FACTORABLE so we must use their graphs to determine the
multiplicity of their roots.
Remember the total number of zeros or roots will equal the degree of P(x).
Even Multiplicity
Roots with even multiplicity touch the
x-axis, but do not cross through it.
Odd Multiplicity
Multiplicity of 1: Crosses “Straight”
through the x-axis.
Multiplicity > 1: Crosses “Bending”
through the x-axis
EXAMPLE:
Identify the roots of 3x5 + 18x4 + 27x3 = 0. State the multiplicity of each root.
Step 1: The total number of roots of this
polynomial will be ______.
Step 2: From the graph we can identify 2 real
roots: x = _____ & x = _____.
Step 3: Look at whether the multiplicity of the
roots is even or odd, then determine their value.
The multiplicity of the root x = _____ is _____
The multiplicity of the root x = _____ is _____
6-5 & 6-6
Finding All Roots of Polynomial Equations
6-5 Example 3: Identifying Multiplicity from a Graph
Identify the roots of each equation. State the multiplicity of each root.
A. x3 + 6x2 + 12x + 8 = 0
Step 1: The total number of roots of this
polynomial will be ______.
Step 2: Identify integer roots from the graph.
Step 3: Look at whether the multiplicity of the
root is even or odd, then determine its value.
The multiplicity of root : _________ is ______
B. x4 + 8x3 + 18x2 – 27 = 0
D. -2x5 +26x3 – 72x = 0
You may want to change your window settings to see the
roots better, or check them in your table. Here is an
example window setting you could try.
C. x4 + 8x3 + 24x2 – 32x + 16 = 0
6-5 & 6-6
Finding All Roots of Polynomial Equations
6-6 Example 2: Writing Polynomial Functions
Write the simplest polynomial with the given roots.
A.
–1, ½, and 4.
Step 1: Write the
equation in factored
form.
Step 2: Multiply.
Conjugate Root Pairs:
If the polynomial P(x) has rational coefficients and
a is a root of P(x), then so
is its conjugate_______.
Likewise, if the polynomial P(x) has rationalcoefficients and bi is a root of P(x),
then so is its conjugate ______.
6-6 Example 3: Writing a Polynomial Function with Complex Zeros
Write the simplest function with the given zeros.
A.
5 and 2i
Step 1: Identify all
roots.
Step 2: Write the
equation in factored
form.
Step 3: Multiply.
6-5 & 6-6
Finding All Roots of Polynomial Equations
6-6 Example 3: Writing a Polynomial Function with Complex Zeros Contd.
Write the simplest function with the given zeros.
B.
2 and
C.
5i
and
3
3