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 left parenthesis 2 x squared plus 3 x plus 1 right parenthesis plus left parenthesis 3 x squared plus 6 x plus 9 right parenthesis

(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 left parenthesis 2 x squared plus 3 x squared right parenthesis plus left parenthesis 6 x plus 3 x right parenthesis plus left parenthesis 1 plus 9 right parenthesis

(2x2+6x+1)+(3x2+3x+9)=−x2+3x−8
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 negative x squared plus 3 x minus 8

(2x2+6x+1)+(3x2+3x+9)=5x2+9x+10

Answer 1

To correctly demonstrate polynomial addition, we need to add the like terms from the two polynomial expressions.

Let's look at the expression \((2x^2 + 6x + 1) + (3x^2 + 3x + 9)\):

1. Combine the \(x^2\) terms: \(2x^2 + 3x^2 = 5x^2\)
2. Combine the \(x\) terms: \(6x + 3x = 9x\)
3. Combine the constant terms: \(1 + 9 = 10\)

Putting these together, we get: \[ (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \]

Now, let's analyze the options given:

1. First Response: \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 13x^2 + 9x + 10 \) - Incorrect (the \(x^2\) terms were added incorrectly)
2. Second Response: \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^4 + 9x^2 + 10 \) - Incorrect (the polynomial degree is wrong)
3. Third Response: \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8 \) - Incorrect (completely wrong values)
4. Fourth Response: \( (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \) - Correct

Therefore, the correct example demonstrating polynomial addition is the fourth response:

\((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = 5x^2 + 9x + 10\).