Multiplicity and End Behavior

What is the multiplicity of an x-intercept?

• The highest exponent of a polynomial

• The number of different factors

• The exponent of the factor corresponding to that x-intercept

• The number in front of a polynomial

Complete the sentence (2 right answers): the multiplicity of an x-intercept...

• Determines exactly what happens near the x-int on the graph

• Helps with the calculation of the x-int

• May determine the sign of the leading coefficient

• Suggests two possibilities for the behavior of the graph near the x-int

Tags

CCSS.HSA.APR.B.3

Given
 $f\left(x\right)\ =\ 10x^2\left(2x-2\right)^3\left(x^2+3\right)$ ,
what is the multiplicity of the x-intercept $x=0$  ?

• 10

• 3

• 2

• 0

Tags

CCSS.HSA.APR.B.3

True or false: an x-intercept can have a multiplicity of  $0$ . Either way, explain your reasoning.

Evaluate responses using AI:

OFF

Tags

CCSS.HSA.APR.B.3

CCSS.HSA.REI.D.10

CCSS.HSA.REI.D.11

Given  $f\left(x\right)\ =\ \left(2x\ +6\right)^7\left(x-3\right)^4$ ,
what is the multiplicity of the x-intercept  $x=-3$  ?

• 7

• 2

• 6

• 4

Tags

CCSS.HSA.APR.B.3

At the root  $x=1$  on  $f\left(x\right)\ =\ \left(3x+3\right)^7\left(6x-6\right)^2\left(x^2+1\right)$  the graph...

• Bounces off of  the x-axis like a parabola

• Passes through the x-axis like a line

• Passes through the x-axis like a "wave"

•  $x=1$  is not a root

Tags

CCSS.HSA.APR.B.3

On the graph of  $f\left(x\right)\ =\ \left(x^2+5\right)\left(x-3\right)^3$ ,  what occurs at  $x\ =\ \sqrt{5}$  ?

• Passes through the x-axis like a line 

• Bounces off of the x-axis like a parabola

• Passes through the x-axis like a cubic

• Does not pass through the x-axis

Tags

CCSS.HSA.APR.B.3

The end behavior of a polynomial is...

• the polynomial's behavior near very large x-intercepts

• the polynomial's behavior far beyond the extremal x-intercepts

• the polynomial's behavior between x-intercepts

• the coefficient of the highest exponent term

Which x-values could you plug in to test the end behavior of 
 $f\left(x\right)\ =\ 100x\left(1000x-10\right)^2\left(x-10\right)$  

•  $x=1$  

•  $x=10$  

•  $x=200$  

•  $x=-10$  

Tags

CCSS.HSA.APR.B.3

CCSS.HSA.CED.A.2

CCSS.HSA.REI.D.10

What is the leading coefficient of  
 $f\left(x\right)=3\left(2x+1\right)^2\left(-x+5\right)^3$ ?

• 12

• 3

• -12

• -3

Tags

CCSS.HSA.APR.A.1