If A = - 2x$^2$ + 12x, B=11-8x+3x$^2$, C=17-4x$^2$ and D = x$^2$-x-3, then what is the sum of the co-efficients of x$^2$ and x in (A + B + C −D) ?

Q: If A = - 2x$^2$ + 12x, B=11-8x+3x$^2$, C=17-4x$^2$ and D = x$^2$-x-3, then what is the sum of the co-efficients of x$^2$ and x in (A + B + C −D) ?

To determine the sum of the coefficients of $x^2$ and $x$ in the expression $A + B + C - D$, we first need to simplify the expression by combining like terms from each polynomial. Given: - $A = -2x^2 + 12x$ - $B = 11 - 8x + 3x^2$ - $C = 17 - 4x^2$ - $D = x^2 - x - 3$ Let's calculate $A + B + C - D$: 1. Combine the $x^2$ terms: \[ (-2x^2 + 3x^2 - 4x^2) - x^2 = -2x^2 + 3x^2 - 4x^2 - x^2 = -4x^2 \] 2. Combine the $x$ terms: \[ (12x - 8x) + x = 12x - 8x + x = 5x \] 3. Combine the constant terms: \[ 11 + 17 - 3 = 25 \] Now, the expression $A + B + C - D$ simplifies to: \[ -4x^2 + 5x + 25 \] Next, we find the sum of the coefficients of $x^2$ and $x$: - Coefficient of $x^2$ is $-4$ - Coefficient of $x$ is $5$ Sum of these coefficients: \[ -4 + 5 = 1 \] Thus, the sum of the coefficients of $x^2$ and $x$ in the expression $A + B + C - D$ is 1. **Explanation of Options:** 1. **Option 1 (0):** Incorrect because the sum of the coefficients $-4 + 5$ is not 0. 2. **Option 2 (-1):** Incorrect because the sum of the coefficients $-4 + 5$ is not -1. 3. **Option 3 (1):** Correct because the sum of the coefficients $-4 + 5$ is indeed 1. 4. **Option 4 (3):** Incorrect because the sum of the coefficients $-4 + 5$ is not 3. Therefore, the correct answer is Option 3, which is 1.

Question

Accepted Answer

To determine the sum of the coefficients of $x^2$ and $x$ in the expression $A + B + C - D$, we first need to simplify the expression by combining like terms from each polynomial.

Given:
- $A = -2x^2 + 12x$
- $B = 11 - 8x + 3x^2$
- $C = 17 - 4x^2$
- $D = x^2 - x - 3$

Let's calculate $A + B + C - D$:

1. Combine the $x^2$ terms:
   $$
   (-2x^2 + 3x^2 - 4x^2) - x^2 = -2x^2 + 3x^2 - 4x^2 - x^2 = -4x^2
   $$

2. Combine the $x$ terms:
   $$
   (12x - 8x) + x = 12x - 8x + x = 5x
   $$

3. Combine the constant terms:
   $$
   11 + 17 - 3 = 25
   $$

Now, the expression $A + B + C - D$ simplifies to:
$$
-4x^2 + 5x + 25
$$

Next, we find the sum of the coefficients of $x^2$ and $x$:
- Coefficient of $x^2$ is $-4$
- Coefficient of $x$ is $5$

Sum of these coefficients:
$$
-4 + 5 = 1
$$

Thus, the sum of the coefficients of $x^2$ and $x$ in the expression $A + B + C - D$ is 1.

**Explanation of Options:**

1. **Option 1 (0):** Incorrect because the sum of the coefficients $-4 + 5$ is not 0.
2. **Option 2 (-1):** Incorrect because the sum of the coefficients $-4 + 5$ is not -1.
3. **Option 3 (1):** Correct because the sum of the coefficients $-4 + 5$ is indeed 1.
4. **Option 4 (3):** Incorrect because the sum of the coefficients $-4 + 5$ is not 3.

Therefore, the correct answer is Option 3, which is 1.