To simplify the expression \((x^2 + 8x - 48)(x^2 - 144) \div (4 - x)\), we will follow these steps:
- Factor each part of the expression.
- Rewrite the expression using the factored form.
- Simplify the resulting expression.
Step 1: Factor the expressions
Factor \(x^2 + 8x - 48\): To factor \(x^2 + 8x - 48\), we need to find two numbers that add to \(8\) and multiply to \(-48\). These two numbers are \(12\) and \(-4\).
So, \[ x^2 + 8x - 48 = (x + 12)(x - 4) \]
Factor \(x^2 - 144\) (a difference of squares): \[ x^2 - 144 = (x - 12)(x + 12) \]
Step 2: Rewrite the entire expression:
Now substitute the factored forms into the original expression: \[ \frac{(x + 12)(x - 4)(x - 12)(x + 12)}{4 - x} \]
Note that \(4 - x = -(x - 4)\), so we can rewrite the expression as: \[ \frac{(x + 12)(x - 4)(x - 12)(x + 12)}{-(x - 4)} \]
Step 3: Simplify the expression:
Cancel out the \((x - 4)\) terms: \[ = - (x + 12)(x + 12)(x - 12) \] \[ = - (x + 12)^2 (x - 12) \]
Final Result:
The expression in the simplest form is: \[ -(x + 12)(x + 12)(x - 12) \]
Similar Form:
Now, based on the responses provided in the question, it looks like the equivalent simplest form could match any of the given options. The first one is: \[ -(x + 4)(x - 4)(x + 12) \] This is not the correct answer.
The second one is: \[ (x - 4)(x + 4)x - 12 \] Also not correct.
The third one is: \[ -\frac{1}{x - 12} \] This seems incorrect because we still have the factor.
The fourth one is: \[ \frac{1}{x - 12} \] which does not match our expression.
Therefore, from the responses provided, we can conclude: None of the suggested forms are matching the simplified expression.
The correct answer should be \( - (x + 12)^2 (x - 12) \), and it appears to not be listed among the choices.
1 answer