Polynomials: Multiplicity

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VOCABULARY: Multiplicity

Multiplicity refers to the number of times a root occurs.

Example: y = (x - 3)(x + 7)²(x - 5)³


x = 3 has a multiplicity of 1
x = -7 has a multiplicity of 2
x = 5 has a multiplicity of 3

Multiple Choice

What does multiplicity mean?

• The number of times a zero occurs.

• The number of factors being multiplied.

• The degree of a polynomial.

Dropdown

Identify the multiplicity of each zero.

f(x) = x⁴ (x - 9)³ (x + 12)



x = 0 has a multiplicity of​ ​

Select answer

x = 9 has a multiplicity of ​

Select answer

x = -12 has a multiplicity of ​

Select answer

Dropdown

Identify the multiplicity of each zero.

f(x) = x² -2x - 24



Hint: Factor f(x) first.



(smallest number) x = ​

Select answer

has a multiplicity of​ ​

Select answer

(largest number) x = ​

Select answer

has a multiplicity of ​

Select answer

Multiple Choice

Imagine the graph of g(x) = (x - 5)²

What happens at x = 5?

• The graph "bounces" off of the x-axis, creating a minimum/maximum at that zero.

• The graph crosses through the x-axis.

Multiple Choice

Imagine the graph of k(x) = -x - 4.

What happens at x = -4?

• The graph "bounces" off of the x-axis, creating a minimum/maximum at that zero.

• The graph crosses through the x-axis.

Multiple Choice

Use your graphing calculator to graph

f(x) = (x - 3)² (x + 4)⁵

What happens at x = 3?

• The graph "bounces" off of the x-axis, creating a minimum/maximum at that zero.

• The graph crosses through the x-axis.

Multiple Choice

Use your graphing calculator to graph

f(x) = (x - 3)² (x + 4)⁵

What happens at x = -4?

• The graph "bounces" off of the x-axis, creating a minimum/maximum at that zero.

• The graph crosses through the x-axis.

On the next slide, observe the patterns in behavior when multiplicity is even and odd.

Match

Without a calculator, match the polynomial with its graph.

f(x)= x³-3x²

f(x)= -2x³+8x

f(x)= -x³+9x

f(x)= 3x⁴-3x³-3x²+3x

f(x)= -2x²+16x-24

• f(x)= x³-3x²

• f(x)= -2x³+8x

• f(x)= -x³+9x

• f(x)= 3x⁴-3x³-3x²+3x

• f(x)= -2x²+16x-24

Match

Without a calculator, match the polynomial with its graph.

f(x)=-3(x-1)(x-2)²(x-3)

f(x)=(x-1)(x-3)(x-5)

f(x)= -(x-4)(x-3)(x-1)²

f(x)=-5

f(x)=x

• f(x)=-3(x-1)(x-2)²(x-3)

• f(x)=(x-1)(x-3)(x-5)

• f(x)= -(x-4)(x-3)(x-1)²

• f(x)=-5

• f(x)=x

Match

Without a calculator, match the polynomial with its graph.

f(x)= 9-4x²

f(x)= x²(x-3)³

f(x)= x⁴-3x³

f(x)= -2(x+3)²(x+1)²

f(x)= x⁴-6x³+8x²

• f(x)= 9-4x²

• f(x)= x²(x-3)³

• f(x)= x⁴-3x³

• f(x)= -2(x+3)²(x+1)²

• f(x)= x⁴-6x³+8x²