Which example correctly demonstrates polynomial subtraction?

(2x2+6x+1)−(3x2+3x+9)=x2+3x+8

(2x2+6x+1)−(3x2+3x+9)=5x2+9x+10

(2x2+6x+1)−(3x2+3x+9)=−x2+3x−8

(2x2+6x+1)−(3x2+3x+9)=−5x2+3x−8

1 answer

Let's perform the polynomial subtraction step-by-step using the expression \((2x^2 + 6x + 1) - (3x^2 + 3x + 9)\):

  1. Distribute the negative sign across the second polynomial: \[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 2x^2 + 6x + 1 - 3x^2 - 3x - 9 \]

  2. Combine like terms:

    • For \(x^2\) terms: \(2x^2 - 3x^2 = -1x^2\) (or \(-x^2\))
    • For \(x\) terms: \(6x - 3x = 3x\)
    • For constant terms: \(1 - 9 = -8\)
  3. Putting it all together, we get: \[ -x^2 + 3x - 8 \]

Now, let's compare this result to the provided examples:

  1. Example 1: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = x^2 + 3x + 8 \) (Incorrect)
  2. Example 2: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10 \) (Incorrect)
  3. Example 3: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8 \) (Correct)
  4. Example 4: \( (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -5x^2 + 3x - 8 \) (Incorrect)

Thus, the correct example demonstrating polynomial subtraction is:

\((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\).