More Key Factors of Polynomials

Recall: From Lesson 4

Standard form (left to right)

Factored form

823 3 xx )2)(3)(6( xxx

The FTA (Fundamental Theorem of Algebra) states that the maximum number of possible solutions to a polynomial equation is

equal to the degree of the polynomial.

If a polynomial is in factored form, you can use the zero product property to find values that will make the polynomial equal zero – or

in other words, find the solution(s)!

Zeros, roots, solutions and x-intercepts are all closely related. They can be written as x = #, #, #

etc. But we can rewrite as ordered pairs.

Multiplicity refers to how many times a factor

appears in the factored form of the polynomial.

Important Points on Graphs

In this lesson we are going to look at the important points that we can find on the graphs of polynomials. The terms we will focus on in this lesson are:

o x-intercepto y-intercepto Vertex/verticeso Axis of Symmetryo Maximumo Minimum

Some polynomials only have one of these points while others could have many or all, and some have multiples! Let’s take a look.

InterceptsYou are very familiar with the x- and y-intercepts and you already know how to find them on a graph of a linear or quadratic function.

Here we see four x-intercepts and one y-intercept. Not all intercepts are easy to identify from a graph because they do not all fall on an integer.

Next we will look at how to find the intercepts.

Intercepts

Intercepts of linear functions: y = mx + b • we know b is the y-intercept - we can also find this by

substituting 0 in for x and solving for y (useful if your equation is not in standard form*)

• we know we can find the x-intercept by substituting 0 in for y and solving for x

Intercepts of quadratic functions: y = ax2 + bx + c • we know c is the y-intercept - we can also find this by

substituting 0 in for x and solving for y (useful if your equation is not in standard form*)

• we know we can find the x-intercept(s) by factoring the quadratic function and solving factors for x to identify the roots, or we can use the quadratic formula or complete the square.

Intercepts

Intercepts of cubic functions: y = ax3 + bx2 + cx + d • we know d is the y-intercept - we can also find this by

substituting 0 in for x and solving for y (useful if your equation is not in standard form*)

• we know we can find the x-intercept by factoring (if the cubic is factorable) the cubic function and solving factors for x

Intercepts of other polynomials:• This is sometimes more difficult but we have methods to find

these. We will explore some strategies later in this lesson. Often you can use a calculator to find the intercepts but using the other key factors of the graph and the concept of multiplicity can be helpful as well!

Intercepts: Quartic ExampleThe quartic function to the right has four x-intercepts and one y-intercept as we have already pointed out.

It appears we can easily identify the y-intercept and two of the four x intercepts easily, but what about the outer two x-intercepts?

34 24 xxy

We can try to factor this function…

Then we can solve to find the roots (x-intercepts)…

*Notice there is no multiplicity here.

34 24 xxy

Intercepts: Quartic Example

1

1

1

01

2

2

2

x

x

x

x

732.13*

3

3

3

03

2

2

2

x

x

x

x

31

3422

24

xxy

xxy

Vertices, Axis of Symmetry, and Max/Min

We also need to investigate the:

o Vertex/verticeso Axis of Symmetryo Maximumo Minimum

The vertex is powerful. If we find the vertex we can identify the axis of symmetry and the min or max.

Again, some polynomials only have a vertex while others could have many!

VertexThe formula we can use to find the vertex of a quadratic function is:

We can easily pull the a and b from the equation to plug in to this formula. Here a = 1 and b = 0

So, we have:

We use x=0 and substitute it back into the function to find the y-value of the ordered pair that represents the vertex.

Vertex: ( 0, - 0.75 )

75.02 xy

a

bx

2

***Remember that when a term is not present in an equation it is because the coefficient is 0. Here we have a 0x term.

75.002 xxy

0)1(2

0x

75.0

75.00

75.00 2

y

y

y

Axis of SymmetryThe axis of symmetry is a vertical line that cuts a graph in half. The x-coordinate of the vertex is the value for the vertical line that is the axis of symmetry. So, if we have

Vertex: ( 0, - 0.75 )

The axis of symmetry is x = 0. 75.02 xy

Let’s graph x2 – 4x – 5 = 0

First we need to find the vertex.

Vertex = (2, -9)

The vertex tells us that the axis of symmetry is x = 2

Then we graph can graph the x-intercepts.

Remember, to find these we need to factor

the polynomial: (x-5)(x+1) x - 5 = 0 x + 1 = 0

x = 5 x = -1

Therefore, the x intercepts are: (5, 0) and (-1, 0)

Example: Graph using Vertex and Intercepts

5242

2

4

)1(2

)4(

22y

a

bx

-9

2

Let’s graph x2 – 4x – 5 = 0

Vertex = (2, -9)

axis of symmetry is x = 2

x-intercepts: (5, 0) and (-1, 0)

Once you have the vertex and the intercepts you can draw the graph for this function.

***Many times you may want to know the y-intercept. Remember that to find the y-intercept we set x = 0 and solve for y.

x2 – 4x – 5 = 0

x2 – 4x – 5 = y

02 – 4(0) – 5 = y

0 – 0 – 5 = y

-5 = y y - intercept is (0, -5)

Example: Graph using Vertex and Intercepts

Let’s graph x2 – 4x – 5 = 0

Vertex = (2, -9)

axis of symmetry is x = 2

x-intercepts: (5, 0) and (-1, 0)

y - intercept is (0, -5)

You can also find more points using your calculator.

Graph the equation and use the table function in your calculator. If you do not know how to do this, ask your teacher or Google it

Example: Graph using Vertex and Intercepts

Let’s graph x2 – 4x – 5 = 0

Vertex = (2, -9)

axis of symmetry is x = 2

x-intercepts: (5, 0) and (-1, 0)

y - intercept is (0, -5)

Finally, let’s determine if there is a minimum or maximum for this graph. Just as the name implies, minimum is the lowest point (vertex) and maximum is the highest point (vertex).

Because we have one vertex on this graph it either has a max or min.

Because the vertex is lower than all the other points on the graph, it is a minimum!

The vertex is a minimum

Example: Graph using Vertex and Intercepts

34 24 xxy

Max/Min for our Quartic ExampleRemember, a minimum is the lowest point (vertex) and a maximum is the highest point (vertex).

Here we have ups and downs – three vertices – so we will have a three max/min’s.

max

minmin