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Which example correctly demonstrates polynomial addition?

A. (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10
B. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -5x^2 + 3x + 8
C. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = x^2 + 3x + 8
D. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8

3 answers

To correctly demonstrate polynomial addition, we need to add the like terms of the given polynomials. Let's look at the addition in choice C:

\[ (2x^2 + 6x + 1) + (3x^2 + 3x + 9) \]

Now, we'll combine the like terms:

  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 it all together, we get:

\[ 5x^2 + 9x + 10 \]

Now we can compare this result with each option:

  • A. \(5x^2 + 9x + 10\) (Correct, but this is a subtraction statement, not an addition).
  • B. \(-5x^2 + 3x + 8\) (Incorrect).
  • C. \(x^2 + 3x + 8\) (Incorrect).
  • D. \(-x^2 + 3x - 8\) (Incorrect).

The correct demonstration of polynomial addition is NOT listed among the choices provided. However, the accurately computed addition is \(5x^2 + 9x + 10\) as shown from the correct calculation steps.

Thus, the answer based solely on polynomial addition should be \(5x^2 + 9x + 10\) but note that none of the choices were correct in demonstrating that operation directly.

Which example correctly demonstrates polynomial subtraction?
A. (2x^2 + 6x + 1) - (3x^2 + 3x + 9) = 5x^2 + 9x + 10
B. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -5x^2 + 3x + 8
C. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = x^2 + 3x + 8
D. (2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8

To correctly demonstrate polynomial subtraction, we need to subtract the second polynomial from the first polynomial. Let's look at the subtraction in option A:

\[ (2x^2 + 6x + 1) - (3x^2 + 3x + 9) \]

Now we'll distribute the negative sign and combine like terms:

  1. Distributing the negative sign gives: \[ (2x^2 + 6x + 1) - 3x^2 - 3x - 9 \]

  2. Combine the \(x^2\) terms: \[ 2x^2 - 3x^2 = -1x^2 \text{ or } -x^2 \]

  3. Combine the \(x\) terms: \[ 6x - 3x = 3x \]

  4. Combine the constant terms: \[ 1 - 9 = -8 \]

Putting it all together, we get:

\[ -x^2 + 3x - 8 \]

Now we can compare this result with the options provided:

  • A. \((2x^2 + 6x + 1) - (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (Correct).
  • B. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -5x^2 + 3x + 8\) (Incorrect).
  • C. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = x^2 + 3x + 8\) (Incorrect).
  • D. \((2x^2 + 6x + 1) + (3x^2 + 3x + 9) = -x^2 + 3x - 8\) (Incorrect).

Therefore, option A correctly demonstrates polynomial subtraction with the result \(-x^2 + 3x - 8\).