Write the quotient and remainder when we divide :
`(x^(2) - 4)` by (x + 2)

Answer

To find the quotient and remainder when dividing \( x^2 - 4 \) by \( x + 2 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We need to divide \( x^2 - 4 \) by \( x + 2 \). We can write this as:

\[ \frac{x^2 - 4}{x + 2} \]

### Step 2: Divide the leading terms
Divide the leading term of the dividend \( x^2 \) by the leading term of the divisor \( x \):

\[ \frac{x^2}{x} = x \]

### Step 3: Multiply and subtract
Now, multiply \( x \) by the entire divisor \( x + 2 \):

\[ x \cdot (x + 2) = x^2 + 2x \]

Now, subtract this result from the original polynomial \( x^2 - 4 \):

\[ (x^2 - 4) - (x^2 + 2x) = -2x - 4 \]

### Step 4: Repeat the process
Now, we need to divide the new leading term \(-2x\) by the leading term of the divisor \(x\):

\[ \frac{-2x}{x} = -2 \]

### Step 5: Multiply and subtract again
Multiply \(-2\) by the entire divisor \(x + 2\):

\[ -2 \cdot (x + 2) = -2x - 4 \]

Now, subtract this from \(-2x - 4\):

\[ (-2x - 4) - (-2x - 4) = 0 \]

### Step 6: Write the final result
Since we have reached a remainder of 0, we conclude that:

- The **quotient** is \( x - 2 \)
- The **remainder** is \( 0 \)

Thus, when dividing \( x^2 - 4 \) by \( x + 2 \), we have:

\[ \text{Quotient} = x - 2, \quad \text{Remainder} = 0 \]

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