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Multiplicity Solutions

Recall that a function’s input (x-value) that makes the output (y-value) equal to zero is called a root or a zero . 
On a graph, the roots are where the function crosses or touches the x-axis. 
Multiplicity is the number of times a root occurs in a polynomial equation. In other words, multiplicity is how many times an x-value makes the equation equal to zero. 
And so, multiplicity determines what a graph will look like when it intersects the x-axis .
- ${(x - h)}^{2}$ is a double root (or a trinomial square).
- ${(x - h)}^{3}$ is a triple root .
Use the leading coefficient test and multiplicities to create a sketch that estimates the zeros of a polynomial.  
When creating a sketch, recall that roots are plotted on the x-axis .

Sketches, unlike exact graphs, do not need to include every detail.
Sketches in this lesson do not include the y-axis.

Multiplicity	Description/Implication	Shorthand	Sketch
1	cross-through	C
2	bounce (also called double root ) Note Any higher even multiplicities will make the point of intersection with the x-axis flatter.	B
3	snake Note Any higher odd multiplicities will make the point of intersection with the x-axis flatter.	S

The descriptions/implications provided in the table are not technical math vocabulary; rather, they provide a way to describe multiplicity in simple language.

To state the possible multiplicities and the degree given a graph:

Perform the leading coefficient test (to estimate a, n). 
Name or estimate the roots on the x-axis . 
Estimate the multiplicity at each root. 
Determine the possible degree n .

Example 1

Write the polynomial as a product of factors using the graph.

Note

Start by determining the possible degree and multiplicities.

a > 0 
n = even degree

C = cross-through 
B = bounce
S = snake

Estimated degree:

$n = 2 + 3 + 2 + 1 = 8$

$a {(x - (- 2))}^{2} {(x - 0)}^{3} {(x - 2)}^{2} (x - 4) = 0$

$a x^{3} {(x + 2)}^{2} {(x - 2)}^{2} (x - 4) = 0 or f (x) = a x^{3} {(x + 2)}^{2} {(x - 2)}^{2} (x - 4)$

Note

Remember, when finding the roots, you set the x-values equal to zero, which means that your equation can either be set equal to zero or f(x).

Example 2

Write the polynomial as a product of factors using the graph. Explain.

a < 0 
n = odd degree

Estimated degree:

n = 3 + 2 = 5

$- a {(x - (- 5))}^{3} {(x - (- 3))}^{2} = 0 - a {(x + 5)}^{3} {(x + 3)}^{2} = 0$

Explain

The end behavior represents a negative odd degree polynomial function. 
There is a triple root ( multiplicity 3 ) at –5 and a double root ( multiplicity 2 ) at –3.

Note

Ask yourself: Based on multiplicity, what is the minimum degree this could be?