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:

• 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)