# Prerequisites

## Polynomials


### Learning Objectives

1. Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.


### Objective 1: Distinguish between a fixed and a growth mindset, and how these ideas may help in learning.

Stanford University psychologist and researcher, Carol Dweck, PH.D., published a book in 2006 called "Mindset, The New Psychology of Success", which changed how many people think about their talents and abilities. Based on decades of research Dr. Dweck outlined two mindsets and their influence on our learning.

Dr. Dweck’s research found that people who believe that their abilities could change through learning and practice (growth mindset) more readily accepted learning challenges and persisted through these challenges.
While individuals who believe that knowledge and abilities come from natural talent and cannot be changed (fixed mindset) more often become discouraged by failure and do not persist. 
Her research shows that if we believe we can learn and master something new, this belief greatly improves our ability to learn.

1. 
Read through the following illustration based on Dr. Dweck’s work.

2. 
It’s important to note that we as individuals do not have a strict fixed or growth mindset at all times. We can lean one way or another in certain situations or when working in different disciplines or areas. For example, a person who often plays video games may feel they can learn any new game that is released and be confident in these abilities, but at the same time avoid sports and are fixed on the idea that they will never excel at physical activities.

In terms of learning new skills in mathematics, which mindset, growth or fixed, best describes your beliefs as of today? Explain.

3. 
Identify each of the following statements as coming from a student with a fixed mindset or with a growth mindset.



4. 
Mindsets can be changed. As Dr. Dweck would say “You have a choice. Mindsets are just beliefs. They are powerful beliefs, but they are something in your mind and you can change your mind.” 
Think about what you would like to achieve in your classes this term and how a growth mindset can help you reach these goals. Write three goals for yourself below.

Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants to find the area of the front of the library so that they can purchase the correct amount of paint. Using the measurements of the front of the house, shown in , we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.
First find the area of the square in square feet.
Then find the area of the triangle in square feet.
Next find the area of the rectangular door in square feet.
The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get 
  or 
  ft2.
In this section, we will examine expressions such as this one, which combine several variable terms.

### Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as 
  is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product 
  such as 
  is a term of a polynomial. If a term does not contain a variable, it is called a constant.
A polynomial containing only one term, such as 
  is called a monomial. A polynomial containing two terms, such as 
  is called a binomial. A polynomial containing three terms, such as 
  is called a trinomial.
We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

### Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, 
  and 
  are like terms, and can be added to get 
  but 
  and 
  are not like terms, and therefore cannot be added.

### Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

### Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the 
  in 
  to obtain the equivalent expression 
  When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

### Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial.
The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

### Perfect Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.
Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.

### Difference of Squares

Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let’s see what happens when we multiply 
  using the FOIL method.
The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few examples.
Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

### Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

### Key Equations


### Key Concepts

1. A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See .
2. We can add and subtract polynomials by combining like terms. See  and .
3. To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See .
4. FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See .
5. Perfect square trinomials and difference of squares are special products. See  and .
6.  Follow the same rules to work with polynomials containing several variables. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, identify the degree of the polynomial.
For the following exercises, find the sum or difference.
For the following exercises, find the product.
For the following exercises, expand the binomial.
For the following exercises, multiply the binomials.
For the following exercises, multiply the polynomials.

### Real-World Applications


### Extensions

For the following exercises, perform the given operations.