James says x+2

Question

James says x+2
 is a factor of x^3-3x+2
  . Susan says he is incorrect and that x+2
 is not a factor of x^3-3x+2
  .

Determine which student is correct and explain how you know by using concepts learned in Unit 7. Also, show any work that is needed to support your explanation.

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GPT-4o mini · Answer

To determine whether $ x + 2 $ is a factor of the polynomial $ x^3 - 3x + 2 $, we can use the Factor Theorem. According to the Factor Theorem, $ x + c $ is a factor of a polynomial $ f(x) $ if and only if $ f(-c) = 0 $. In this case, since we have $ x + 2 $, we set $ c = -2 $. Now, we will evaluate the polynomial $ f(x) = x^3 - 3x + 2 $ at $ x = -2 $: $$ f(-2) = (-2)^3 - 3(-2) + 2 $$ $$ = -8 + 6 + 2 $$ $$ = -8 + 8 $$ $$ = 0 $$ Since $ f(-2) = 0 $, according to the Factor Theorem, $ x + 2 $ is indeed a factor of the polynomial $ x^3 - 3x + 2 $. Thus, James is correct, and Susan is incorrect. The work shows that $ x + 2 $ divides $ x^3 - 3x + 2 $ without a remainder, confirming it is a factor.