# Prerequisites

## Factoring Polynomials


### Learning Objectives:

1. Master a proven technique for note taking: The Cornell Method.
2. Use the Cornell Process to study from your notes.

### Objective 1: Master a proven technique for note taking: The Cornell Method.

The Cornell Method for taking notes was created by an education professor at Cornell University, Dr. Walter Pauk. The Cornell Method consists of two strategies, a format for notetaking and then the process of using your notes to study. The format for note taking involves separating out the pages of your notebook into three or four separate regions. This method can help summarize information from a lecture, a video, or a reading from your text. You don’t need to purchase Cornell paper, just divide your sheets into regions like the illustration below. Write on just one side of the paper so that you can later fold back the left column and quiz yourself using practice test questions.

The top portion is called the heading. This is where you write your name, the class, the date, the section of the text and a main topic objective for the day.

The right section is where you will take notes during class. Try to summarize main ideas here without copying word for word everything your teacher is saying. Use bullet points or numbers to prioritize important ideas. Include here the definitions your teacher presents, and the examples worked in class. It is useful if you can group the content here by learning objectives.
The left column about 2 inches wide is used to write questions about the main concepts that were covered in class. For example, write sample test questions over the concepts discussed in class in this column. This is also where you should write cues about the importance of the information including vocabulary terms, diagrams, formulas or the methods being used to solve. If you like to sketch this is a place to add illustrations for key ideas or icons so you can quickly identify certain components.

The summary will be written in the lower section about 2 inches from the bottom. You will complete this after class summarizing the important concepts you were taught in a short compact way. Think of this summary as what you might describe to a friend who happened to miss class that day. Reflect on the important ideas here. Add illustrations or a mind map to your summary if this helps. You will use this section to find information later when studying.


### Practice Makes Perfect


### Objective 2: Use the Cornell Process to study from your notes.


The Cornell Way. Using your Cornell notes to study is referred to as the Cornell Way. 
The process includes Cornell note taking (presented in Objective 1), note making, note interacting, and note reflecting. 
After taking notes in class, follow up with these study methods.

Note making: This is where you will fill in any gaps left in your Cornell notes meaning add in any missed details. Fill in your question column with sample test questions, any formulas used, and highlight, circle or star important any important ideas. Complete your summary row with a few sentences summarizing the important ideas. This should be completed right after class or within the next day. Be creative and add some color, make these notes into something you enjoy working with.

Note interacting: This is the ongoing process of studying from your notes. Fold your left question column back and ask yourself the practice test questions. Include note interactions in the review for your exams. These note interactions can be as short as 5 minutes in length but need to happen regularly and at least daily for the week before exams.

Note reflecting: This is where you assess how helpful your notes were. Do this right after you get back your graded exam. Did the regular note interactions help you to perform better on the exam? Were there problems similar to those you predicted on the exam? Enhance your notes with any important ideas you had initially left out. Then use what you learn from this assessment to improve your note taking in the future.


### Practice Makes Perfect

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in .
The area of the entire region can be found using the formula for the area of a rectangle.
The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of 
  units2. The other rectangular region has one side of length 
  and one side of length 
  giving an area of 
  units2. So the region that must be subtracted has an area of 
  units2.
The area of the region that requires grass seed is found by subtracting 
  units2. This area can also be expressed in factored form as 
  units2. We can confirm that this is an equivalent expression by multiplying.
Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

### Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 
  is the GCF of 
  and 
  because it is the largest number that divides evenly into both 
  and 
  The GCF of polynomials works the same way: 
  is the GCF of 
  and 
  because it is the largest polynomial that divides evenly into both 
  and 
 

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

### Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial 
  has a GCF of 1, but it can be written as the product of the factors 
  and 
 

Trinomials of the form 
  can be factored by finding two numbers with a product of 
  and a sum of 
  The trinomial 
  for example, can be factored using the numbers 
  and 
  because the product of those numbers is 
  and their sum is 
  The trinomial can be rewritten as the product of 
  and 
 


### Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial 
  can be rewritten as 
  using this process. We begin by rewriting the original expression as 
  and then factor each portion of the expression to obtain 
  We then pull out the GCF of 
  to find the factored expression.

###  Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.
We can use this equation to factor any perfect square trinomial.

### Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.
We can use this equation to factor any differences of squares.

### Factoring the Sum and Difference of Cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.
Similarly, the difference of cubes can be factored into a binomial and a trinomial, but with different signs.
We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.
The sign of the first 2 is the same as the sign between 
  The sign of the 
  term is opposite the sign between 
  And the sign of the last term, 4, is always positive.

### Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance, 
  can be factored by pulling out 
  and being rewritten as 
 


### Key Equations

1. The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See .
2. Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See .
3. Trinomials can be factored using a process called factoring by grouping. See .
4. Perfect square trinomials and the difference of squares are special products and can be factored using equations. See  and .
5. The sum of cubes and the difference of cubes can be factored using equations. See  and .
6. Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the greatest common factor.
For the following exercises, factor by grouping.
For the following exercises, factor the polynomial.
For the following exercises, factor the polynomials.

### Real-World Applications

For the following exercises, consider this scenario:
Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area of 
  m2, as shown in the figure below. The length and width of the park are perfect factors of the area.
For the following exercise, consider the following scenario:
A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below.  The flagpole will take up a square plot with area 
  yd2.

### Extensions

For the following exercises, factor the polynomials completely.