Find the sum of `4x^(3)- 3x^(2) +2x- 5 " and " -7x^(3) + 6x^(2) - x + 11`.

Options

A. `3x^(3) - 3x^(2) + x + 6`
B. `-3x^(3) +3x^(2) + x + 6`
C. `3x^3 + 3x^(2) - x -6`
D. `3x^(3) - 3x^(2)-x +6`

Answer

To find the sum of the two algebraic expressions \(4x^3 - 3x^2 + 2x - 5\) and \(-7x^3 + 6x^2 - x + 11\), we will follow these steps:

### Step 1: Write down the expressions
We start with the two expressions:
1. \(4x^3 - 3x^2 + 2x - 5\)
2. \(-7x^3 + 6x^2 - x + 11\)

### Step 2: Combine the expressions
We will add the two expressions together:
\[ (4x^3 - 3x^2 + 2x - 5) + (-7x^3 + 6x^2 - x + 11) \]

### Step 3: Remove the parentheses
This simplifies to:
\[ 4x^3 - 3x^2 + 2x - 5 - 7x^3 + 6x^2 - x + 11 \]

### Step 4: Group like terms
Now, we will group the like terms together:
- Cubic terms: \(4x^3 - 7x^3\)
- Quadratic terms: \(-3x^2 + 6x^2\)
- Linear terms: \(2x - x\)
- Constant terms: \(-5 + 11\)

### Step 5: Simplify each group
Now we will simplify each group:
1. For cubic terms: \(4x^3 - 7x^3 = -3x^3\)
2. For quadratic terms: \(-3x^2 + 6x^2 = 3x^2\)
3. For linear terms: \(2x - x = 1x\) or simply \(x\)
4. For constant terms: \(-5 + 11 = 6\)

### Step 6: Write the final expression
Combining all the simplified terms, we get:
\[ -3x^3 + 3x^2 + x + 6 \]

### Final Answer
Thus, the sum of the two expressions is:
\[ -3x^3 + 3x^2 + x + 6 \]

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