Properties of Polynomials

Properties of Polynomials

End Behavior, Roots, and Equations

End Behavior

• The way a function continues on BOTH ENDS of the function.

End Behavior

• Top Left: Both sides are continuing upward, so both endbehaviors are POSITIVE

• Bottom Left: Both sides are continuing Downward so both endbehaviors are NEGATIVE

• Top Right: On the negative side of the x-axis (x --> neg inf) the graph continues downward (y --> neg inf) On the positive side of the x-axis (x --> pos inf), the graph continues upward (y --> pos inf)

• Bottom Right: As x --> neg inf y --> pos inf (continues upward) As x --> pos inf, y --> neg inf (continues downward)

Multiple Choice

Which image represents the description:

 $As\ \ \ x\ \rightarrow\ \ -\infty,\ \ \ y\ \rightarrow\ \ -\infty$   $As\ \ \ x\ \rightarrow\ \ \infty,\ \ \ y\ \rightarrow\ \ -\infty$  

Click on the image to enlarge

• Graph I

• Graph II

• Graph III

• Graph IV

Multiple Choice

Which image represents the description:

 $As\ \ \ x\ \rightarrow\ \ -\infty,\ \ \ y\ \rightarrow\ \ \infty$   $As\ \ \ x\ \rightarrow\ \ \infty,\ \ \ y\ \rightarrow\ \ -\infty$  

Click on the image to enlarge

• Graph I

• Graph II

• Graph III

• Graph IV

Multiple Choice

Which image represents the description:

 $As\ \ \ x\ \rightarrow\ \ -\infty,\ \ \ y\ \rightarrow\ \ \infty$   $As\ \ \ x\ \rightarrow\ \ \infty,\ \ \ y\ \rightarrow\ \ \infty$  

Click on the image to enlarge

• Graph I

• Graph II

• Graph III

• Graph IV

Multiple Choice

Which image represents the description:

 $As\ \ \ x\ \rightarrow\ \ -\infty,\ \ \ y\ \rightarrow\ \ -\infty$   $As\ \ \ x\ \rightarrow\ \ \infty,\ \ \ y\ \rightarrow\ \ \infty$  

Click on the image to enlarge

• Graph I

• Graph II

• Graph III

• Graph IV

Number & Types of Roots

Number is based on the degree (aka highest exponent value)

Type is based on how many interesections in the x-axis

Number

• Given the equation:

   $y=x^5+4x^2-2x-3$  

• There are FIVE roots (because the highest exponentn is "5"

Type

• Given the equation:

   $y=x^5+4x^2-2x-3$  

• There are THREE REAL roots b/c the function passes through the x-axis THREE times

• With the degree of "FIVE" that means that there are TWO remaining IMAGINARY roots

Multiple Choice

Given the equation and graph, determine the number and types of roots

 $y=2x^3\ +x^2-4x$ 

• 2 real roots, no imaginary

• 3 real roots, 1 imaginary

• 2 real roots, 2 imaginary

• 3 real roots, no imaginary

Multiple Choice

Given the equation and graph, determine the number and types of roots

 $y=\ \ -x^4+3x^2-2x-4$ 

• 2 real roots, 2 imaginary

• 3 real roots, 1 imaginary

• 2 real roots, 1 imaginary

• 4 real roots, no imaginary

Multiple Choice

Given the equation and graph, determine the number and types of roots

 $y=\ \ -x^4+2x-1$ 

• 2 real roots, 2 imaginary

• 3 real roots, 1 imaginary

• 2 real roots, 1 imaginary

• 4 real roots, no imaginary

Equations

Using End Behavior and Zeros to build equations

Equations

• The end behavior (on the right) is continuing upward, so the coefficient will be POSITIVE

• The ZEROS are -3, 1, and 2

• The equation would be

• y = + (x + 3)(x - 1)(x - 2)

• (Remember Factors Flip)

Equations

• The end behavior (on the right) is continuing downward, so the coefficient will be NEGATIVE

• The ZEROS are -1 (bounce = twice), 0, and 2

• The equation would be

• y = — x (x + 1)² (x − 2)

• (Remember Factors Flip)

Multiple Choice

Is the leading coefficient positive or negative

• Positive

  based on the LEFT side of the graph

• Positive

  based on the RIGHT side of the graph

• Negative

  based on the LEFT side of the graph

• Negative

  based on the RIGHT side of the graph

Multiple Choice

Using the graph, which factor will have a power of 2?

• (x + 3)²

• (x + 1)²

• (x + 120)²

• (x − 4)²

Multiple Choice

Given the image, determine the CORRECT way to write the equation

(click on image to enlarge)

**Be careful of COEFFICIENT

• $y=(x−3)(x+1)(x+2)$

• $y=\ -(x−3)(x+1)(x+2)$
  

• $y=(x+3)(x−1)(x−2)$
  

• $y=\ \ -(x+3)(x−1)(x−2)$
  

Multiple Choice

Given the image, determine the CORRECT way to write the equation

(click on image to enlarge)

**Be careful of COEFFICIENT

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

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

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

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

Multiple Choice

Given the image, determine the CORRECT way to write the equation

(click on image to enlarge)

**Be careful of COEFFICIENT

•  $y=\left(x-a\right)^2\left(x+b\right)$ 

•  $y=\ -\left(x-a\right)^2\left(x+b\right)$ 
  

•  $y=\left(x+a\right)\left(x-b\right)^2$ 
  

•  $y=\ -\left(x+a\right)\left(x-b\right)^2$ 
  

Multiple Choice

Determine which polynomial would have the zeros −3, 5,−i, i

Reminder

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