The sum of `x^(3) + x^(2) - 2x, 5x - 2x^(2) + 7x^(3) " and " 8x^(2) - 10x + 10x^(3)` is

Options

A. `18x^(3) + 7x^(2) - 7x`
B. `11x^(3) - 10x^(2) - 5x`
C. `18x^(3) - 13x^(2)- 7x`
D. `10x^(3) - 11x^(2) - 5x`

Answer

To find the sum of the algebraic expressions \( x^3 + x^2 - 2x \), \( 5x - 2x^2 + 7x^3 \), and \( 8x^2 - 10x + 10x^3 \), we will follow these steps:

### Step 1: Write the expressions in a single equation
We start by writing the expressions together, replacing the commas with plus signs:
\[ (x^3 + x^2 - 2x) + (5x - 2x^2 + 7x^3) + (8x^2 - 10x + 10x^3) \]

### Step 2: Remove the parentheses
Since we are adding, we can remove the parentheses without changing the signs:
\[ x^3 + x^2 - 2x + 5x - 2x^2 + 7x^3 + 8x^2 - 10x + 10x^3 \]

### Step 3: Combine like terms
Now we will combine the like terms. Like terms are terms that have the same variable raised to the same power.

1. **Combine \( x^3 \) terms:**
- \( x^3 + 7x^3 + 10x^3 = (1 + 7 + 10)x^3 = 18x^3 \)

2. **Combine \( x^2 \) terms:**
- \( x^2 - 2x^2 + 8x^2 = (1 - 2 + 8)x^2 = 7x^2 \)

3. **Combine \( x \) terms:**
- \( -2x + 5x - 10x = (-2 + 5 - 10)x = -7x \)

### Step 4: Write the final expression
Putting it all together, we have:
\[ 18x^3 + 7x^2 - 7x \]

### Final Answer:
The sum of the given expressions is:
\[ \boxed{18x^3 + 7x^2 - 7x} \]

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