# 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 QuadrantDown Arrow: 3rd Quadrant Up Arrows: 1st and 2nd Quadrant Leading Coefficient: Negative Up Arrow: 2nd QuadrantDown 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)

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