Writing and Graphing Polynomial Equations

Julie Wagner: Rockledge High School

Algebra 2, Advanced Topics, Pre-Calc, Trig/Ananlt Geometry

Wagner.julie@brevardschools.org

Objective: To graph, analyze, write, and describe polynomial equations with real and complex roots.

NGSSS: MA.912.A.4.5 Graph polynomial functions with and without

technology and describe end behavior. MA.912.A.4.6 Use the Fundamental Theorem of Algebra. MA.912.A.4.7 Write a polynomial equation for a given set of

real and/or complex roots.

Key Concepts:

Roots of a polynomial Describe end behavior Write an equation in standard form from

given roots Rational Root Theorem Find the zeros of a polynomial equation Imaginary Roots

EXAMPLE: Graph a fourth-degree polynomial with four real roots.

Roots: -6, -2, 1, 5

Using your roots (zeros) write a polynomial function in standard form.

x = -6, -2, 1, 5 F(x) = (x + 6)(x + 2)(x – 1)(x – 5) F(x) = (x2 + 8x + 12)(x2 – 6x + 5) F(x) = x4 – 6x3 + 5x2 + 8x3 – 48x2 + 40x + 12x2 - 72x + 60 F(x) = x4 + 2x3 – 31x2 – 32x + 60

Since the right side behavior opens down, we need

our leading coefficient negative. Multiply by -1. F(x) = –x4 – 2x3 + 31x2 + 32x – 60

Now lets check our work.

Work backwards to find the roots F(x) = -x4 – 2x3 + 31x2 + 32x – 60

Use the Rational Root Theorem to list all the possible rational roots.

±1 ±2 ±3 ±4 ±5 ±6 ±10 ±12 ±15± 20 ±30 ±60

1. Use the Rational Root Theorem. ±1 ±2 ±3 ±4 ±5 ±6 ±10 ±12 ±15± 20 ±30 ±60

2. Use synthetic division with x = 1. 1 -1 -2 31 32 -60 -1 -3 28 60 -1 -3 28 60 0

P(x) = (x – 1)(-x3 – 3x2 + 28x + 60)

3. Use synthetic division with x = -2.

-2 -1 -3 28 60

2 2 -60

-1 -1 30 0

F(x) = (x – 1)(x + 2)(-x2 – x + 30)

F(x) = (x – 1)(x + 2)(-x + 5)(x + 6)

Roots: x = 1, -2, 5, -6

The roots should be the same as when you started.

DIRECTIONS:

Get into groups of 2-4 people. Grab 2 string, scissors, glue bottle, & 4

sheets of graph paper & cut along dotted line. Cut string into 2 equal length pieces.

Label the X and Y-Axis. Take 20-30 minutes to construct third and

fourth degree polynomial graphs fitting the described roots. Answer the questions that follow.

THE END!!!

Wagner.julie@brevardschools.org Julie Wagner Rockledge High School 321-636-3711 x262