6-5 & 6-6 Finding All Roots of Polynomial Equations Warm Up: Factor each expression completely. 1. 2y 3 + 4y 2 – x 4 – 6x 2 – : Use factoring to solve polynomial equations : Use a graph and synthetic division to identify all real roots of a polynomial equation : 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. 4x 6 + 4x 5 – 24x 4 = 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. x = 26x 2 C. x 3 – 2x 2 – 25x = –50 A.Identify all the real roots of the polynomial equation. 2x 3 – 3x 2 –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 divide the integer you found in Step 1 into 2x 3 – 3x 2 –10x – 4. 2 – 3 –10 – Example 2: Identify All of the Real Roots of a Polynomial Equation

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. 2x 3 – 9x = 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 divide the 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. x 4 – 3x 3 + 5x 2 – 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 divide the integer you found in Step 1 into _____________________.

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. x 4 + 4x 3 – x x – 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 divide the integer you found in Step 1 into _____________________.

6-5 & 6-6 Finding All Roots of Polynomial Equations 6.5.3: Define and identify multiplicity of roots of polynomial functions in factored form or from a graph : Write a polynomial function given all rational zeros : 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:3x x x 3 = 0 FACTORED FORM:3x 3 (x + 3) (x + 3) = 0 OR 3x 3 (x + 3) 2 = 0 ZERO PRODUCT PROPERTY:3x 3 = 0 (x + 3) = 0 ROOTS: Degree of Polynomial: ________ Degree of factor 3x 3 : _________Degree of factor (x + 3): ______ Multiplicity of root x = _____;______

6-5 & 6-6 Finding 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 MultiplicityOdd 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. 2x 6 – 10x 5 – 12x 4 = 0 Identify the roots of each equation. State the multiplicity of each root. B. x 4 – 13x = 0 EXAMPLE: Identify the roots of 3x x x 3 = 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 _____

6-5 & 6-6 Finding 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. x 3 + 6x x + 8 = 0 B. x 4 + 8x x 2 – 27 = 0 D. -2x 5 +26x 3 – 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. x 4 + 8x x 2 – 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 & 6-6 Finding All Roots of Polynomial Equations Write the simplest polynomial with the given roots. A. –1, ½, and 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 ______.

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. C.