Which example correctly demonstrates polynomial addition?(1 point)

Question

Responses

(2x2+6x+1)+(3x2+3x+9)=13x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 13 x squared plus 9 x plus 10

(2x2+6x+1)+(3x2+3x+9)= 5x4+9x2+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 5 x quartic plus 9 x squared plus 10

(2x2+6x+1)+(3x2+3x+9)=5x2+9x+10
left parenthesis 2 x squared plus 6 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 3 x plus 9 right parenthesis equals 5 x squared plus 9 x plus 10

(2x2+6x+1)+(3x2+3x+9)=−x2+3x−8

Answer 1

To correctly demonstrate polynomial addition, you need to combine like terms from the two polynomial expressions being added together. Let's evaluate the options given.

1. Option 1:
   \[(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10\]

   • Combine the like terms:
     • \(2x^2 + 3x^2 = 5x^2\)
     • \(6x + 3x = 9x\)
     • \(1 + 9 = 10\)
   • This results in \(5x^2 + 9x + 10\), which does not equal \(13x^2 + 9x + 10\).
   • Incorrect.

2. Option 2:
   \[(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10\]

   • This option introduces an incorrect \(x^4\) term.
   • Incorrect.

3. Option 3:
   \[(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\]

   • As calculated previously:
     • \(2x^2 + 3x^2 = 5x^2\)
     • \(6x + 3x = 9x\)
     • \(1 + 9 = 10\)
   • This result is indeed correct: \(5x^2 + 9x + 10\).
   • Correct.

4. Option 4:
   \[(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8\]

   • This result is incorrect because it does not represent a proper addition of the original polynomials.
   • Incorrect.

The only option that correctly demonstrates polynomial addition is Option 3:
\[(2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\]