Multiplicity — Algebra II, what it means and where it is taught

How many times a root repeats. It decides whether the graph crosses or just touches.

Taught in Algebra II.

Where it is taught

Practise this idea

3 questions in the course bank use Multiplicity, in Polynomial Functions & Their Graphs.

Problem 1

At x=3, what does the graph of y=(x-3)2(x+1) do?

  1. Touches and turns back
  2. Crosses straight through
  3. Has a hole
  4. Is undefined
Worked answer

Touches and turns back

(x-3) appears twice, so multiplicity 2

Even multiplicity bounces off the axis

Problem 2

At x=-1, what does y=(x-3)2(x+1) do?

  1. Crosses straight through
  2. Touches and turns back
  3. Has a hole
  4. Flattens out
Worked answer

Crosses straight through

Multiplicity 1 is odd

Odd multiplicity crosses

Problem 3

Write the factored polynomial of least degree that touches at x=2 and crosses at x=-3, leading coefficient 1.

Worked answer

(x-2)^2(x+3)

Touch at 2 means (x-2)2

Cross at -3 means (x+3)

Elsewhere

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