Divide :
`x^(2) + 3x - 54` by x - 6

Answer

To divide the polynomial \( x^2 + 3x - 54 \) by \( x - 6 \), we can follow these steps:

### Step 1: Factor the polynomial \( x^2 + 3x - 54 \)

We need to express the quadratic polynomial in factored form. We are looking for two numbers that multiply to \(-54\) (the constant term) and add up to \(3\) (the coefficient of \(x\)).

The two numbers that satisfy these conditions are \(9\) and \(-6\) because:
- \(9 \times (-6) = -54\)
- \(9 + (-6) = 3\)

Thus, we can factor the polynomial as:
\[ x^2 + 3x - 54 = (x + 9)(x - 6) \]

### Step 2: Set up the division

Now, we can rewrite the division of the polynomial by \(x - 6\):
\[ \frac{x^2 + 3x - 54}{x - 6} = \frac{(x + 9)(x - 6)}{x - 6} \]

### Step 3: Cancel the common factor

Since \(x - 6\) is a common factor in the numerator and the denominator, we can cancel it out:
\[ \frac{(x + 9)(x - 6)}{x - 6} = x + 9 \]

### Step 4: Write the final answer

Thus, the result of the division is:
\[ \boxed{x + 9} \]

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