1. Essential Questions How do we solve equations and inequalities involving polynomials? Standards MM3A3: Students will solve a variety of equations and inequalities. MM3A3a: Find real and complex roots of higher degree polynomial equations using the factor theorem, remainder theorem, rational root theorem, and fundamental theorem of algebra, incorporating complex and radical conjugates. MM3A3b: Solve polynomial, exponential, and logarithmic equations analytically, graphically, and using appropriate technology. MM3A3c: Solve polynomial, exponential, and logarithmic inequalities analytically, graphically, and using appropriate technology. Represent solution sets of inequalities using interval notation.

2. Activating Strategy Which of the following is a factor of the polynomial x 4 – 12x 3 + 52x 2 – 96x + 64? A) (x – 3)B) (x – 8)C) (x + 2)D) (x – 2)

3. The “solutions” of the equation are the same as the “roots” of the polynomial since we are being asked to find the x-values that make the function equal 0. x 4 – 12x 3 + 52x 2 – 96x + 64 = 0

4. Solve the equations by graphing the left side of the equation as one function and the right side of the equation as a second function, then finding the x- coordinates at the intersection point(s). x 3 + 5x 2 - x = 5 Solutions: -5, -1, 1 x 4 – 2x 3 – 13x 2 + 18x + 13 = 4x – 11 Solutions: -3, -1, 2, 4 Notice: The solutions are only the x coordinates.

5. Sketch the graphs, label the ordered pairs at the intersection point(s), and state the solutions for the equations. 1. x 3 + 7x 2 = -7x + 15 2. 2x 4 – 5x 3 + 5x 2 = 20x + 12 3. x 3 – 8x 2 + 8x + 3 = 67 4. x 4 – 10x 2 + 6 = 2x + 6