Investigating Polynomial Behavior - Multiplicities

Investigating Polynomial Behavior - Multiplicities

 $2x^4\left(x-1\right)$  

• Factors =  $2,\ x^4,\ and\ \left(x-1\right)$  

• Set each factor with a variable equal to 0

•  $Solve\ x^4=0$  

•  $x=0\ with\ multiplicity\ 4$  .... The graph BOUNCES off 0

•  $Solve\ x-1=0$  

•  $x=1\ with\ multiplicity\ 1$  ... The graph CROSSES at 1

 $-6x\left(x+1\right)^5$  

• Factors =  $-6,\ x,\ and\ \left(x+1\right)^5$  

• Set each factor with a variable equal to 0

•  $Solve\ x=0$  

•  $x=0\ multiplicity\ 1$  .... The graph CROSSES at 0

•  $Solve\ x+1=0$  

•  $x=-1\ multiplicity\ 5$  ... The graph WIGGLES at -1

 $3x\left(x-1\right)^3\left(x+1\right)^2$  

• Factors =  $3,\ x,\ \left(x-1\right)^3,\ and\ \left(x+1\right)^2$  

•  $Solve\ x=0$  

•  $x=0\ multiplicity\ 1$  .... The graph CROSSES at 0

•  $Solve\ x-1=0$  

•  $x=1\ multiplicity\ 3$  ... The graph WIGGLES at 1

•  $Solve\ x+1=0$  

•  $x=-1\ multiplicity\ 2$  ... The graph BOUNCES at -1

Breakout Exploration

•  $y=-5x\left(x-1\right)^6\left(x+1\right)^7$  

•  $y=-5x^2\left(x+1\right)^3\left(x-1\right)^4$  

•  $y=4x^2\left(x-1\right)^5\left(x+1\right)$  

•  $200x^2\left(x-1\right)^5\left(x+1\right)^6$  

Multiple Choice

A polynomial function has a zero at 4 (multiplicity 3) and 0 (multiplicity 1). Write a function that could represent this function.

• $f\left(x\right)=x\left(x-4\right)^3$

• $f\left(x\right)=x\left(x+4\right)^3$

• $f\left(x\right)=\left(x-4\right)^3$

• $f\left(x\right)=x^3\left(x-4\right)^4$

Multiple Choice

What is the effect on the graph if a zero has a multiplicity of 4?

• cross

• bounce

• wiggle (change direction)

Multiple Choice

What is the effect on the graph if a zero has a multiplicity of 1?

• cross

• bounce

• wiggle (change direction)

Multiple Choice

What is the effect on the graph if a zero has a multiplicity of 5?

• bounce

• cross

• wiggle (change direction)