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# Writing and graphing polynomials

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### Writing and graphing polynomials

1. 1. Writing and Graphing Polynomial Equations Julie Wagner: Rockledge High School Algebra 2, Advanced Topics, Pre-Calc, Trig/Ananlt Geometry [email_address]
2. 2. Objective: To graph, analyze, write, and describe polynomial equations with real and complex roots. <ul><li>NGSSS: </li></ul><ul><li>MA.912.A.4.5 Graph polynomial functions with and without technology and describe end behavior. </li></ul><ul><li>MA.912.A.4.6 Use the Fundamental Theorem of Algebra. </li></ul><ul><li>MA.912.A.4.7 Write a polynomial equation for a given set of real and/or complex roots. </li></ul>
3. 3. Key Concepts: <ul><li>Roots of a polynomial </li></ul><ul><li>Describe end behavior </li></ul><ul><li>Write an equation in standard form from given roots </li></ul><ul><li>Rational Root Theorem </li></ul><ul><li>Find the zeros of a polynomial equation </li></ul><ul><li>Imaginary Roots </li></ul>
4. 4. EXAMPLE: Graph a fourth-degree polynomial with four real roots. <ul><li>Roots: -6, -2, 1, 5 </li></ul>
5. 5. Using your roots (zeros) write a polynomial function in standard form. <ul><li>x = -6, -2, 1, 5 </li></ul><ul><li>F(x) = (x + 6)(x + 2)(x – 1)(x – 5) </li></ul><ul><li>F(x) = (x 2 + 8x + 12)(x 2 – 6x + 5) </li></ul><ul><li>F(x) = x 4 – 6x 3 + 5x 2 + 8x 3 – 48x 2 + 40x + 12x 2 - 72x + 60 </li></ul><ul><li>F(x) = x 4 + 2x 3 – 31x 2 – 32x + 60 </li></ul><ul><li>Since the right side behavior opens down, we need </li></ul><ul><li>our leading coefficient negative. Multiply by -1. </li></ul><ul><li>F(x) = –x 4 – 2x 3 + 31x 2 + 32x – 60 </li></ul>
6. 6. Now lets check our work. <ul><li>Work backwards to find the roots </li></ul><ul><li>F(x) = -x 4 – 2x 3 + 31x 2 + 32x – 60 </li></ul><ul><li>Use the Rational Root Theorem to list all the possible rational roots. </li></ul><ul><li>±1 ±2 ±3 ±4 ±5 ±6 ±10 ±12 ±15± 20 ±30 ±60 </li></ul>
7. 7. <ul><li>1. Use the Rational Root Theorem. </li></ul><ul><li>±1 ±2 ±3 ±4 ±5 ±6 ±10 ±12 ±15± 20 ±30 ±60 </li></ul><ul><li>Use synthetic division with x = 1. </li></ul><ul><li>1 -1 -2 31 32 -60 </li></ul><ul><li>-1 -3 28 60 </li></ul><ul><li>-1 -3 28 60 0 </li></ul><ul><li>P(x) = (x – 1)(-x 3 – 3x 2 + 28x + 60) </li></ul>
8. 8. 3. Use synthetic division with x = -2. <ul><li>-2 -1 -3 28 60 </li></ul><ul><li>2 2 -60 </li></ul><ul><li>-1 -1 30 0 </li></ul><ul><li>F(x) = (x – 1)(x + 2)(-x 2 – x + 30) </li></ul><ul><li>F(x) = (x – 1)(x + 2)(-x + 5)(x + 6) </li></ul><ul><li>Roots: x = 1, -2, 5, -6 </li></ul>
9. 9. <ul><li>The roots should be the same as </li></ul><ul><li>when you started. </li></ul>
10. 10. DIRECTIONS: <ul><li>Get into groups of 2-4 people. </li></ul><ul><li>Grab 2 string, scissors, glue bottle, & 4 sheets of graph paper & cut along dotted line. Cut string into 2 equal length pieces. </li></ul><ul><li>Label the X and Y-Axis. </li></ul><ul><li>Take 20-30 minutes to construct third and fourth degree polynomial graphs fitting the described roots. Answer the questions that follow. </li></ul>
11. 11. THE END!!! <ul><li>[email_address] </li></ul><ul><li>Julie Wagner </li></ul><ul><li>Rockledge High School </li></ul><ul><li>321-636-3711 x262 </li></ul>