Analyzing zeros of polynomial graphs

Analyzing zeros of polynomial graphs

by Karine Ptak

​PART 1

​MULTIPLICITY

​YOU SHOULD WRITE THIS DOWN...

Multiple Choice

If we follow the definition of multiplicity,  $$\left(x-1\right)^3=0$$  would result in  $$x=1$$  with a multiplicity of...

• 0

• 1

• 2

• 3

​Keep writing...

​Keep writing...

​Before we start... We use the following terms almost interchangeably. Although there are some distinctions, they represent more or less the same thing:

solutions, zeros, roots, and x-intercepts​

Multiple Choice

What kind of multiplicity would the zeros of this graph have?

• x=-1 has an odd multiplicity

  x=2 has an odd multiplicity

• x=-1 has an odd multiplicity

  x=2 has an even multiplicity

• x=-1 has an even multiplicity

  x=2 has an odd multiplicity

• x=-1 has an even multiplicity

  x=2 has an even multiplicity

​PART 2

​The truth about imaginary zeros

​When you learned about the discriminant...

• ​We can use the discrimant to help us figure out the number and the nature of roots/zeros/solutions

IMAGINARY ROOTS/ZEROS...

... ALWAYS COME IN PAIRS

​PART 3

​The Fundamental Theorem of Algebra

​The number of zeros of a polynomial function is equal to the degree of the polynomial function.

​Make sure you understand the sentence above.

​What that means:

Linear equation: degree is 1, one solution

​Quadratic equation: degree is 2, two solutions

​Cubic equation: degree is 3, three solutions

​

​Get it? ​So, given that the graph on the right only has real solutions, what is the degree of the function?

​These are the graphs of...

​... quadratic functions. Each has two zeros. The green graph has two zeros; they just happen to be imaginary. The blue graph has two zeros, at x=-4 and x=3. The red graph has a zero at x=10, but its multiplicity is 2, so it counts as 2.

​

​

​This is the graph of...

​This is the graph of...

​This is the graph of...

​This is the graph of...

Fill in the Blanks

If you wanted to create a function from this graph that include imaginary solutions, what could be the  degree of your function?