6-5 & 6-6Finding All Roots of Polynomial Equations

Warm Up: Factor each expression completely.1. 2y3 + 4y2 – 30 2. 3x4 – 6x2 – 24

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.

LEARNING GOALS – LESSON 6.5 & 6.6 DAY 1

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 _________ ____________

________________.

Solve the polynomial equation by factoring.

6-5 Example 1: Using Factoring to Solve Polynomial Equations

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 Squares3 Terms: Easy Method or AC Method4 Terms: Grouping

4. Check each factored binomial for Difference of Squares

5. Use the Zero Product Property for roots

6-5 & 6-6Finding 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

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 3 Solve the resulting equation,

_________________________= 0 to find the remaining roots.

Step 2 Use synthetic division to dividethe integer you found in Step 1 into 2x3 – 3x2 –10x – 4.

2 – 3 –10 – 4

6-5 Example 2: Identify All of the Real Roots of a Polynomial Equation

6-5 & 6-6Finding 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 3 Solve the resulting equation,

_________________________= 0 to find the remaining roots.

Step 2: Use synthetic division to dividethe integer you found in Step 1 into _____________________.

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 3 Solve the resulting equation,

_________________________= 0 to find the remaining roots.

Step 2: Use synthetic division to dividethe integer you found in Step 1 into _____________________.

6-5 & 6-6Finding 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 3 Solve the resulting equation,

_________________________= 0 to find the remaining roots.

Step 2: Use synthetic division to dividethe integer you found in Step 1 into _____________________.

6-5 & 6-6Finding All Roots of Polynomial Equations

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.

LEARNING GOALS – LESSON 6.5 & 6.6 DAY 2

THE FUNDAMENTAL THEOREM OF ALGEBRA

Every polynomial function of degree n of 1 or more has EXACTLY n zeros including multiplicity.

Warm-up: Fill in each of the statements to make them true using the given information.

GIVEN: If 5 is a root of P(x) = 0 then . . .

1.) P(5) = ______ 2.) 5 si an ____- intercept of the graph of P(x)

3.) (x - ___) is a factor 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)

Sometimes a polynomial equation has a factor that appears more than once.

This creates a ______________ root.

FACTOR: 3x5 + 18x4 + 27x3 = 0FACTORED FORM: 3x3 (x + 3) (x + 3) = 0 OR 3x3 (x + 3)2 = 0ZERO PRODUCT PROPERTY: 3x3 = 0 (x + 3) = 0ROOTS:

Degree of Polynomial: ________

Degree of factor 3x3: _________ Degree of factor (x + 3): ______

Multiplicity of root x = _____;______ Multiplicity of root x = _____;______

6-5 & 6-6Finding All Roots of Polynomial Equations

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 Odd Multiplicity

Roots with even multiplicity touch the x-axis, but do not cross through it.

Multiplicity of 1: Crosses “Straight” through the x-axis.

Multiplicity > 1: Crosses “Bending” through the x-axis

A. 2x6 – 10x5 – 12x4 = 0

Identify the roots of each equation. State the multiplicity of each root.

B. x 4 – 13x2 + 36 = 0

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 & 6-6Finding All Roots of Polynomial Equations

Identify the roots of each equation. State the multiplicity of each root.

6-5 Example 3: Identifying Multiplicity from a Graph

A. x3 + 6x2 + 12x + 8 = 0

B. x4 + 8x3 + 18x2 – 27 = 0

D. -2x5 +26x3 – 72x = 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 ______

C. x4 + 8x3 + 24x2 – 32x + 16 = 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.

6-5 & 6-6Finding All Roots of Polynomial Equations

Write the simplest polynomial with the given roots.

A. –1, ½, and 4.

6-6 Example 2: Writing Polynomial Functions

Step 1: Write the equation in factored form.

Step 2: Multiply.

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.

Conjugate Root Pairs:

If the polynomial P(x) has rational coefficients and 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 ______.

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a

6-5 & 6-6Finding 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.

C.

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2 and 3

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5i and 3