Find special products of binomials (advanced)

The  exercise appears under the Algebra II Math Mission. This exercise practices finding special products (perfect squares and difference of squares) of "advanced" binomials: binomials with leading coefficients other than 1; binomials with higher degrees; and binomials with two variables.

Types of Problems
There are two types of problems in this exercise:
 * 1) Find the equation has the same solution as ___.: This problem has a binomial equation. The student is asked to find which other equation is equivalent to it.Find special products of binomials (advanced).PNG
 * 2) Express the area of the entire rectangle below as a trinomial.: This problem has an illustration of a rectangle. The student is asked to find the area of the entire figure an type it in the answer box as a trinomial. Find special products of binomials (advanced)2.PNG

Strategies
Knowledge of binomials are encouraged to ensure success on this exercise, but a working knowledge of the Binomial Theorem or multiplication and patience are all that is necessary.
 * 1) The expression on the left-hand side of the equation is a product of two binomials. Students can replace the two terms with their product and obtain an equivalent equation (which means it will have the same solutions).
 * 2) The area of any rectangle is the product of its two dimensions, which we can refer to as length and width. This rectangle is a square as it's length and width are both $$8r^2+1$$. Therefore, the area of this square is $$(8r^2+1)(8r^2+1)$$.

Real-life Applications

 * 1) The binomial theorem has many applications in combinatorics as a counting strategy.
 * 2) The binomial coefficients are related to Pascal's triangle which has many useful properties in number theory and applications in fractal geometry.