Polynomial functionsUnit 1

Multiplicity of a polynomial root

Use a root’s multiplicity to explain how the graph meets the -axis.

  • A factor , with no further factor of , gives the root multiplicity .
  • An odd multiplicity crosses the -axis. An even multiplicity touches the axis and turns back.
  • A multiplicity greater than 1 flattens the graph at the root. Compare the same two roots below.

Example 1

State the roots and their multiplicities, and describe the graph at each root.

Polynomial with roots -1 and 1. At -1, multiplicity 3, crosses. At 1, multiplicity 4, touches and turns.−11
multiplicity 3, crosses.multiplicity 4, touches and turns.

Example 2

State the roots and their multiplicities, and describe the graph at each root.

Polynomial with roots -1 and 1. At -1, multiplicity 4, touches and turns. At 1, multiplicity 3, crosses.−11
multiplicity 4, touches and turns.multiplicity 3, crosses.

Remember
The roots stay at −1 and 1 in both sketches. Swapping multiplicities 3 and 4 swaps which root crosses and which root touches the axis.

Your turn

5 practice questions
Question 1

State the multiplicity of the zero and describe how the graph behaves there.

Check answer
Multiplicity 3; crosses the -axis.
Question 2

State the multiplicity of the zero and describe how the graph behaves there.

Check answer
Multiplicity 2; touches the -axis and turns.
Question 3

State the multiplicity of the zero and describe how the graph behaves there.

Check answer
Multiplicity 4; touches the -axis and turns.
Question 4

State the multiplicity of the zero and describe how the graph behaves there.

Check answer
Multiplicity 2; touches the -axis and turns.
Question 5

State the multiplicity of the zero and describe how the graph behaves there.

Check answer
Multiplicity 5; crosses the -axis.
← All Polynomial functions cardsReport a mistake