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: OddDegree: Even
Leading Coefficient: PositiveUp Arrow: 1st Quadrant
Down Arrow: 3rd Quadrant
Up Arrows: 1st and 2nd Quadrant
Leading Coefficient: NegativeUp 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)


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