# Mathematics: Multiplicity and Curve Sketching

## Roots of multiplicity r

- A root of a polynomial is a value of \(x\) for which \(P(x)=0\)
- For example, \(P(x)=x^2+6x+9\) can be expressed as \((x+3)^2\)
- In this case, \(-3\) is a root of multiplicity 2 of \(P(x)\)

- Roots of multiplicity 1 are also known as “single roots”
- Roots of multiplicity 2 are also known as “double roots”
- Roots of multiplicity 2 are also known as “triple roots”

## Curve Sketching

- Graphs with multiple roots have specific rules for sketching

#### Rules for Leading Coefficient and Degrees

- The table below explains what happens as a graph approaches infinity and negative infinity, based on the leading coefficient and degree:

Degree: Odd | Degree: Even | |

Leading Coefficient: Positive | Up Arrow: 1st Quadrant Down Arrow: 3rd Quadrant | Up Arrows: 1st and 2nd Quadrant |

Leading Coefficient: Negative | Up Arrow: 2nd Quadrant Down Arrow: 4th Quadrant | Down Arrows: 3rd and 4th Quadrant |

- When a graph approaches the x-axis to mark a single root, the x axis is said to be “cut” by the graph (red line)
- When a graph approaches the x-axis to mark an odd multiple root, the graph is said to “slide” at the x-axis (green line)
- When a graph approaches the x-axis to mark an even multiple root, the graph is said to “bounce” at the x-axis (blue line)