Question
Express in simplest form: x2+8x−48x2−144 ÷(4−x) (1 point)
Responses
−(x+4)(x−4)(x+12)
the fraction with numerator negative open paren x plus 4 close paren and denominator open paren x minus 4 close paren times open paren x plus 12 close paren
(x−4)(x+4)x−12
the fraction with numerator open paren x minus 4 close paren times open paren x plus 4 close paren and denominator x minus 12
−1x−12
the fraction with numerator negative 1 and denominator x minus 12
1x−12
Responses
−(x+4)(x−4)(x+12)
the fraction with numerator negative open paren x plus 4 close paren and denominator open paren x minus 4 close paren times open paren x plus 12 close paren
(x−4)(x+4)x−12
the fraction with numerator open paren x minus 4 close paren times open paren x plus 4 close paren and denominator x minus 12
−1x−12
the fraction with numerator negative 1 and denominator x minus 12
1x−12
Answers
GPT-4o mini
To simplify the expression \((x^2 + 8x - 48)(x^2 - 144) \div (4 - x)\), we will follow these steps:
1. Factor each part of the expression.
2. Rewrite the expression using the factored form.
3. 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. Factor each part of the expression.
2. Rewrite the expression using the factored form.
3. 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.