Factor polynomials with special product forms

The  exercise appears under the Algebra I Math Mission. This exercise practices factoring "advanced" polynomials (i.e. polynomials of various degrees and or with two variables) using special product factorization methods.

Types of Problems
There is one type of problem in this exercise:


 * 1) Find the special product form, then find the equivalent expression: This problem provides a quadratic expression. The student is expected to find the special product form, then find the equivalent expression with the same special product form. Students should note that $$a$$ and $$b$$ represent either positive rational numbers or terms with positive coefficients.Factor polynomials with special product forms.PNG

Strategies
Knowledge of any quadratic factoring technique can be used to ensure accuracy and efficiency on this exercise.
 * 1) One such process for factoring a quadratic such as this is to call the coefficient to the $$x$$ the "adding" term and the product of the constant and the leading coefficient to be called the "multiplying" term. Then find two numbers that add to the "adding" number and multiply to the "multiplying" number.
 * 2) Another possible technique is to use the quadratic formula to find the roots and use these to determine the factoring.

Real-life Applications

 * 1) Polynomials can be used via power series to represent many complicated functions and calculus is simpler to perform on polynomials than other functions. Thus any comfort with polynomials should increase calculus ability.
 * 2) Factoring polynomials is an important skill in solving inequalities and equations with polynomials.
 * 3) Factoring polynomials can be used for solving equations with polynomials, such as those that appear in many business models.