Module 3: Lesson 1

 

Self-Check 4

2.  
   

   4. f(x) = (x − 2)(x + 2)(x − 1)(x + 1)(x − 3)

       

      Step 1: Determine the end behaviour by looking at the degree of the polynomial and the sign of the leading coefficient.
      
      There are five factors containing x; therefore, the expanded form of the polynomial is of degree 5. The degree of the polynomial is odd and the polynomial has a positive leading coefficient. This means the graph will start in quadrant 3 and end in quadrant 1.

       

      Step 2: Factor the polynomial if it is not given in the problem.
      
      This question is already factored.

       

      Step 3: Determine the x-intercepts by looking at the factors of the polynomial.
      
      The x-intercepts are x = 2, −2, 1, −1, and 3.

       

      Step 4: Determine the y-intercept by substituting zero into the function.
      
      The y-intercept occurs at the point (0, −12).

       

      Step 5: Determine the nature of the x-intercepts (whether there is a sign change) by looking at the multiplicity of the polynomial’s factors.
      
      Each factor has a multiplicity of 1, so the function will change signs at the corresponding x-intercepts.

       

      Step 6: Draw a smooth curve through the x- and y-intercepts.
      
      

       

      
      

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