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# Why It Matters: Power and Polynomial Functions

You’re suddenly rethinking your decision as you find yourself 456 feet above the ground on the world’s tallest roller coaster—Kingda Ka at Six Flags Great Adventure.  Perhaps you don’t have time to think about it though since before you can catch your breath, you’re already plummeting down in a 270-degree spiral. Within your 50-second ride, you travel over 3,000 feet and reach a top speed of 128 mph. This roller coaster is not for the faint of heart! So how does a thrilling roller coaster experience come to be?  It all begins with advanced planning. Roller coaster designers use the knowledge of math, specifically polynomials, to create an experience that meets specific requirements. Imagine you’re a roller coaster designer entrusted with the task of designing the next big attraction for a nearby theme park. Some criteria for your coaster includes a starting height of 200 ft, dive below ground level at 3 seconds into the ride, return to ground level at 5 seconds, and another dive beneath the ground 10 seconds later. Where do you begin?  Read on because you’ll find the answer to this question and many more in this module. At the end of the module, we’ll return to your design problem so you can come up with a plan.

### Learning Outcomes

Review Topics for Success
• Determine if a given function is a polynomial function.
• Determine the degree and leading coefficient of a polynomial function.
• Identify graphs of polynomial functions.
• Identify general characteristics of a polynomial function from its graph.

Characteristics of Power and Polynomial Functions

• Identify power functions.
• Describe end behavior of power functions given its equation or graph.
• Identify polynomial functions.
• Identify the degree and leading coefficient of polynomial functions.
• Describe the end behavior of a polynomial function.
• Identify turning points of a polynomial function from its graph.
• Identify the number of turning points and intercepts of a polynomial function from its degree.
• Determine x and y-intercepts of a polynomial function given its equation in factored form.

Graphs of Polynomial Functions

• Identify zeros of polynomial functions with even and odd multiplicity.
• Draw the graph of a polynomial function using end behavior, turning points, intercepts, and the Intermediate Value Theorem.
• Write the equation of a polynomial function given its graph.

Dividing Polynomials

• Use long division to divide polynomials.
• Use synthetic division to divide polynomials.

Methods for Finding Zeros of Polynomials

• Evaluate a polynomial using the Remainder Theorem.
• Use the Rational Zero Theorem to find rational zeros.
• Use the Factor Theorem to solve a polynomial equation.
• Use synthetic division to find the zeros of a polynomial function.
• Use the Fundamental Theorem of Algebra to find complex zeros of a polynomial function.
• Use the Linear Factorization Theorem to find polynomials with given zeros.
• Use Descartes’ Rule of Signs to determine the maximum number of possible real zeros of a polynomial function.
• Solve real-world applications of polynomial equations.