Polynomial and Factored Form
Unit 1: Introduction to Polynomial Functions
Activity 4: Factor and Remainder Theorem
Content
In the last activity, you practiced the sketching of a polynomial graph, if you were given the Factored Form of the function statement. In this activity, you will learn a process for developing the Factored Form of a polynomial function, if given the General Form of the function.
Review
A polynomial function is a function whose equation can be expressed in the form of: f(x) = anxn + an-1xn-1 + an-2xn-2 + ...a2x2 + a1x + a0 where an, an-1, ...a1, a0 are real numbers and n is a natural number. The coefficient, which is attached to the highest degree of “x”, is called the leading coefficient.
A polynomial function expressed in the form f(x)= anxn + an-1xn-1 + an-2xn-2 + ...a2x2 + a1x + a0 is said to be in General Form.
Suppose you are given f(x) = anxn + an-1xn-1 + an-2xn-2 + ... a2x2 + a1x + a0 (this is a General Form of a polynomial function).
Your task is to sketch a graph of this function. Since you know how to sketch a graph of a polynomial function,if the function is in Factored Form, it would be handy for you to know how to move easily from General Form to Factored Form.
Tools
Before we continue with this task, you need some tools to help you in the process.
Dividing Polynomials.
Work through the following animation, to introduce you to the idea of dividing polynomials. You will learn two ways to do this! One is called Long Division, and the second is called Synthetic Division. Watch to the clip titled Dividing Polynomials... but please be aware, that, depending on your Internet connection speed, the clip may take a few minutes to download. You can always continue reading the remainder of this page while you wait.
Finding the first factor.
Observation:
In the animation you just viewed, the examples illustrated how to divide a function, f(x), by a factor of the form (jx - k).
Each of the solutions ended with a Division Statement. The
Activity 4: Factor and Remainder Theorem
Content
In the last activity, you practiced the sketching of a polynomial graph, if you were given the Factored Form of the function statement. In this activity, you will learn a process for developing the Factored Form of a polynomial function, if given the General Form of the function.
Review
A polynomial function is a function whose equation can be expressed in the form of: f(x) = anxn + an-1xn-1 + an-2xn-2 + ...a2x2 + a1x + a0 where an, an-1, ...a1, a0 are real numbers and n is a natural number. The coefficient, which is attached to the highest degree of “x”, is called the leading coefficient.
A polynomial function expressed in the form f(x)= anxn + an-1xn-1 + an-2xn-2 + ...a2x2 + a1x + a0 is said to be in General Form.
Suppose you are given f(x) = anxn + an-1xn-1 + an-2xn-2 + ... a2x2 + a1x + a0 (this is a General Form of a polynomial function).
Your task is to sketch a graph of this function. Since you know how to sketch a graph of a polynomial function,if the function is in Factored Form, it would be handy for you to know how to move easily from General Form to Factored Form.
Tools
Before we continue with this task, you need some tools to help you in the process.
Dividing Polynomials.
Work through the following animation, to introduce you to the idea of dividing polynomials. You will learn two ways to do this! One is called Long Division, and the second is called Synthetic Division. Watch to the clip titled Dividing Polynomials... but please be aware, that, depending on your Internet connection speed, the clip may take a few minutes to download. You can always continue reading the remainder of this page while you wait.
Finding the first factor.
Observation:
In the animation you just viewed, the examples illustrated how to divide a function, f(x), by a factor of the form (jx - k).
Each of the solutions ended with a Division Statement. The