Respuesta :

JonHenderson55

Answer:   " x = -2, -1 , 0, and 2 " .

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The zeros of the function:  " f(x) = -x(x+2)(x-2)(x+1) " ;  are:

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           →    " x = -2, -1 , 0, and 2 " .

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Step-by-step explanation:

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We are asked to find the "zeros" of the function:

           →    "  f(x) = -x(x+2)(x-2)(x+1) " ;

            →  that is;  'find all values for "x"  when " f(x) = 0 " ;

So;  let's rewrite this function as follows:

   

           →   f(x) = 0 =  -x(x+2)(x-2)(x+1) ;

             →  0 =  -x(x+2)(x-2)(x+1) ;

So, on the right-hand side of the equation, we have 4 (Four) "multiplicands" ;

  1)  -x ;

  2)  (x+2)  ;

  3)  (x-2) ;

  4)  (x+1) .

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 Since any value, multiplied by "0" ; results in a value of "zero" ;

note that when any of these 4 (Four)  multiplicands is equal to "0" {zero"} ; the entire equation will be equal to "0" {"zero"]" —  (i.e the equation will, in fact, be an equation — both sides of the equation will be equal} ; specifically, the equation as stated; that is, equal to "0" on the "left-hand side" as it is written above.

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So, start with the first multiplicand:

 1)  -x ;   When " -x = 0 " ;  the equation equals "0" ;

        →  " -x = 0 " ;  

             →  Rewrite as:  "  -1x = 0 " ;

  Divide each side of the equation by "-1" ;

    to isolate "x" on one side of the equation;

               & to solve for "x" ;

              →    - 1x / -1  = 0 / - 1   ;

              →   x = 0 .

  (or; by inspection:

              " -x = 0 ;  what is "x" ? ;

              →   " -(?) = 0 " ;  ? = 0 ;  → x = 0 ;  since:  "-0 =0 " ;

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Now, continue with the second multiplicand:

 2)  (x+2) ;   When " (x + 2) = 0 " ;  the equation equals "0" ;

         What is "x" when:  " (x + 2) = 0 "  ?  ;

              →   x + 2 = 0 ;

  Subtract "2" from each side of the equation;

    to isolate "x" on one side of the equation;

               & to solve for "x" ;

              →   x + 2 = 0 ;

              →   x + 2 - 2 = 0 - 2 ;

    to get:

              →  x = - 2 .

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Next, continue with the third multiplicand:

 3)  (x−2) ;   When " (x − 2) = 0 " ;  the equation equals "0" ;

         What is "x" when:  " (x − 2) = 0 "  ?  ;

              →   x − 2 = 0 ;

  Add "2" to each side of the equation;

    to isolate "x" on one side of the equation;

               & to solve for "x" ;

              →   x − 2 = 0 ;

              →   x − 2 + 2 = 0 + 2  ;

    to get:

              →  x = 2 .

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Then, continue with the last and final multiplicand:

  4)  (x+1) ;   When " (x + 1) = 0 " ;  the equation equals "0" ;

         What is "x" when:  " (x + 1) = 0 "  ?  ;

              →   x + 1 = 0 ;

  Subtract "1" from each side of the equation;

    to isolate "x" on one side of the equation;

               & to solve for "x" ;

              →   x + 1 = 0 ;

              →   x + 1 - 1 = 0 - 1 ;

    to get:

              →  x = - 1 .

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The zeros of the function:  "  f(x) = -x(x+2)(x-2)(x+1) "  ;  are:

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           →    " x = -2, -1 , 0 , and 2 " .

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Hope this answer —with explanation— is helpful to you!

    Wishing you the best in your academic endeavors

              — and within the "Brainly" community!

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