This game is fast-paced and perfect for a more competitive class. Divide the class into two groups. Give each group a pile of index cards. Half the cards are polynomials that need to be simplified; the other half are their simplified counterparts. When the game begins, each team tries to quickly match up its polynomials. The first team to do this wins.
This game also uses teams and index cards, but in a different way. Each student on a team gets an index card with a polynomial on it. Two team members pair up and set their polynomials equal to each other. They then solve the resulting equation. For example, if the two students have "5x + 1" and "x^2 - 2" on their index cards, they would have to solve the equation 5x + 1 = x^2 - 2. The first team to have all its pairs solved wins.
Students are often confused when they first learn about the FOIL (first, outer, inner, last) method, which is a technique to multiply two polynomials, for example, (5x + 1)(x^2 - 2). To help them understand how the method works, divide the students into groups and let them design FOIL posters that illustrate the technique. Students can use arrows to show how to multiply the first term of each polynomial together, then the outer terms, then the inner terms, and then the last terms.
In the example, the FOIL technique would give: (5x)(x^2) + (5x)(-2) + (1)(x^2) + (1)(-2).
Advanced students can use graphing calculators to learn more about how polynomials work. Give students a large polynomial, such as .02x^6 + .05x^5 + 3x^4 + 10x^2 - 20,000x + 2000, and have them graph it on their graphing calculators. Tell them the graph represents a roller coaster, and they should work in groups to tweak the polynomial in order to make the first drop more steep, the highest point much higher, or the last hill much lower.