Factoring Polynomials

Factoring Polynomials

A polynomial is the term used to describe graphs of functions that are of degree two or higher. For example:

  1. y = ax2
  2. y = ax3
  3. y = ax4
  4. y = ax5
  5. ...and so on.

The basic shape of each function's graph depends on the exponent of the variable. The exponent describes the maximum number of times the graph crosses the x-axis.

  1. y = ax2 crosses the x-axis 2 times
  2. y = ax3 crosses the x-axis 3 times
  3. y = ax4 crosses the x-axis 4 times
  4. y = ax5 crosses the x-axis 5 times
  5. ...and so on.

Let's look at the graph of each function:

  1. y = ax2
    x^2
  2. y = ax3
    x^3
  3. y = ax4
    x^4
  4. y = ax5
    x^5
Each time a graph crosses the x-axis the point in which it intersects the axis is called an x-intercept of the equation. Each of the factors are represented using the formula (x - b) where b is the x-intercept.
Each x-intercept generates a factor of the equation. A graph that crosses the x-axis 2 times will have a function f(x) = (x - b1)(x - b2). A graph that crosses the x-axis three times will have the function f(x) = (x - b1)(x - b2)(x - b3). A graph that crosses the x-axis 4 times will have a function f(x) = (x - b1)(x - b2)(x - b3)(x - b4) and so on.

Let's look at a few examples.
Given the graph of a function, state the x-intercepts.
ex
The x-intercepts have been highlighted in red. This graph has x-intercepts of b1 = -7, b2 = 3, b3 = 6. Substituting the points into the function f(x) = (x - b1)(x - b2)(x - b3) gives us the following: f(x) = (x + 7)(x - 3)(x - 6). Notice when substituting a negative value for b, it will turn the subtract sign in to a plus sign: (x - (-7)) is equal to (x + 7). When substituting positive values for b, the subtract sign remains unchanged: (x - (3)) equals (x - 3) and (x - (6)) equals (x - 6).

Given the graph, state the x-intercepts from left to right.
ex

Given the graph, state the x-intercepts from left to right.
ex

Given the graph of a function, fill in the equation of the graph.
p6

f(x) = (x - )(x - )(x - )

Given the graph of a function, fill in the equation of the graph.
p8

f(x) = (x - )(x - )(x - )

Given the graph of a function, fill in the equation of the graph.
p9

f(x) = (x - )(x - )(x - )(x - )(x - )