Which polynomial represents the sum below? (4x2 + 3x + 4) + (-3x6 - 4x5 - 2x2 + 5x) a. 3x6 + 4x5 - 2x2 - 8x + 4 b. 3x6 - 4x5 + 2x2 + 8x + 4 c. -x6 - 4x5 + 2x2 + 8x + 4 d. -3x6 - 4x5 + 2x2 + 8x + 4

The correct answer is:  [D]:  "  -3x⁶ − 4x⁵ + 2x² + 8x + 4 " .

Explanation:

Given:


   " ( 4x² + 3x + 4)  +  ( -3x⁶  −  4x⁵  −  2x²  +  5x) " ; 

Rewrite as:  

   " ( 4x² + 3x + 4)  +  1( -3x⁶  −  4x⁵  −  2x²  +  5x) "  ;

Note:  The "1" is implied — since "1", multiplied by any value, is equal to that same value.

Note the "distributive property of multiplication" :


a(b + c)  =   ab + ac ;

a(b − c)  =   ab − ac ;

So, consider the following portion of the expression:

  " + 1 ( -3x⁶  −  4x⁵  −  2x²  +  5x) "  ; 

=   (+1 * -3x⁶)  − (1 * 4x⁵)  − (1 * 2x²)  + (1 * 5x)

=   − 3x⁶ − 4x⁵ − 2x² + 5x ; 

Now, bring down the rest of the expression:

( 4x² + 3x + 4) − 3x⁶ − 4x⁵ − 2x² + 5x ;

   =  4x² + 3x + 4 − 3x⁶ − 4x⁵ − 2x² + 5x

Combine the "like terms" ; 

   + 4x² − 2x²  = + 2x²  ;

   + 3x  +  5x  = 8x  ; 

And we have:
 
   "  -3x⁶ − 4x⁵ + 2x² + 8x + 4 " ; 

   → which is:  Answer choice:  [D]:  "  -3x⁶ − 4x⁵ + 2x² + 8x + 4 " .