March 31, 2021

Chapters

## Polynomial Function

To understand the meaning of a rational function, you should first understand what a polynomial is. A polynomial function is a function made up of **monomials** that have variables. The following are examples of monomials.

If we **break down** each monomial, we can see clearly that they are made up of the following.

A | Constant |

B | Variable |

C, D | Constant and 1 or more variables |

A **polynomial** is made up of monomials. Take a look at some of the examples below.

These **four polynomials** can be classified based on the number of monomials they contain.

A | Monomial | One monomial |

B | Binomial | Two monomials |

C | Trinomial | Three monomials |

D | Polynomial | 4 or more monomials |

## Rational Function Definition

A rational function is any polynomial function divided by another polynomial function. The reason behind the name **‘rational function’** is the term ‘ratio,’ which is simply a fraction.

The image above is an example of a ratio, which can be written in **fraction form.** In practice, any polynomial can be a rational function, the reason why is explained in the image below.

As you can see, any polynomial can be **divided** by the monomial 1. Here are some more examples of rational functions.

Numerator | Denominator | |

Binomial | Binomial | |

Trinomial | Monomial | |

Constant | Constant |

## Linear Function

A linear function is a special type of polynomial. As the name implies, a linear function is any function that involves a linear equation. A linear equation is any **equation** that results in a linear relationship.

y | Output |

m | Slope |

x | Input |

b | Y-intercept |

Some **examples** of linear equations can be seen below.

A | |

B | |

C |

## Rational Function Rules

There are many useful rules that you should know for **rational functions.** Let’s take a look at the rules for linear functions first. Here are the rules you should follow in order to find the slope, y-intercept and any point on a line.

Slope from two points | |

Y-intercept | b = mx = y |

Point on a line | Plug in any x to get a point (x,y) |

In terms of polynomials, we have several **special** polynomials that you should be familiar with.

Quadratic | |

Cubic | |

Circle | |

Ellipse |

One of the most important things you will do when working with rational functions is simplify the polynomials that make them up. A **rational** function can be written as follows.

Where **P(x)** is one polynomial and **Q(x)** is another. The first step you should take is to **simplify** each polynomial. Then, look for like terms. There are two shortcuts that are extremely important. The first is when you have to factor a polynomial that has a perfect square.

As you can see, the trinomial can be simplified in either direction. For a **cubic polynomial,** what out for ones that follow this pattern.

## Problem 1

Classify the following functions as either rational or **non-rational** functions. Justify why you think each function is either rational or non-rational. If they are rational, say what kind of polynomials they are.

1 | f(x) = |

2 | f(x) = |

3 | f(x) = |

4 | f(x) = |

## Problem 2

You are given the following polynomial. First, **simplify** this polynomial. Next, state what kind of polynomial it is and graph it.

## Problem 3

Simplify the following polynomial, making sure to include all the **steps** you took to simplify this polynomial.

## Problem 4

Simplify the following polynomial, making sure to **include** all the steps you took to simplify this polynomial.

## Problem 5

Simplify the following rational **function,** making sure to include all the steps you took to simplify this polynomial.

## Solution Problem 1

In this problem, you were asked to classify the following functions as **either rational** or non-rational functions. You were also tasked with saying what kind of polynomials they are. The answer to this problem can be found below.

1 | f(x) = | Non-rational: includes an imaginary number | N/A |

2 | f(x) = | Rational: constant | Monomial |

3 | f(x) = | Rational: two polynomials | Two monomials |

4 | f(x) = | Rational: two polynomials | Two binomials |

## Solution Problem 2

In this problem you were given the following polynomial. After simplifying this polynomial, you were asked to graph it. The first step we should take is to simplify the x term. Notice that it is the square root of a square, which always gives us the **number** itself.

Now, we just **distribute** the 2 across the binomial.

This is clearly a linear equation. **Plotting** this linear equation, we get the following:

x | y |

-2 | 0 |

-1 | 2 |

0 | 4 |

1 | 6 |

2 | 8 |

3 | 10 |

## Solution Problem 3

In this problem you were asked to **simplify** the following polynomial.

First, we want to write the formula.

Next, we **multiply** each of the terms with each other.

Or, we can **simply** use the rule:

## Solution Problem 4

Here is how you simplify the following polynomial.

First, you should be able to recognize this type of pattern. Because we’re multiplying a binomial and a trinomial, we should always check if we can use a rule. Luckily, we can use the **rule** from before, as we’re dealing with multiples of 3.

## Solution Problem 5

For this rational function, we can combine what we’ve learned to simplify the **numerator** and **denominator.**