Question
To simplify the given expression \( \frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2)} \), we first need to simplify the numerator and the denominator.
### Step 1: Simplify the Numerator
Start with the expression in the numerator:
\[
3x^2 + 2x - 5x - 2x^2 + 2
\]
Combine like terms:
\[
(3x^2 - 2x^2) + (2x - 5x) + 2
\]
This results in:
\[
x^2 - 3x + 2
\]
### Step 2: Factor the Numerator
Now, we will factor \( x^2 - 3x + 2 \):
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]
### Step 3: Simplify the Denominator
Now, let's look at the denominator:
\[
(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2) = (x - 2)^2(x - 1)(x + 2)^2(x + 1)
\]
### Step 4: Rewrite the Complete Expression
Now substituting in the factored numerator gives:
\[
\frac{(x - 1)(x - 2)}{(x - 2)^2(x - 1)(x + 2)^2(x + 1)}
\]
### Step 5: Cancel Common Factors
In this expression, we can cancel out \( (x - 1) \) and one \( (x - 2) \):
\[
\frac{\cancel{(x - 1)} (x - 2)}{(x - 2) \cancel{(x - 1)} (x + 2)^2 (x + 1)} = \frac{1}{(x - 2)(x + 2)^2(x + 1)}
\]
### Final Result:
Thus, the expression that is equivalent to the given one is:
\[
\boxed{\frac{1}{(x - 2)(x + 2)^2(x
make this like a 7th grader wrote it
### Step 1: Simplify the Numerator
Start with the expression in the numerator:
\[
3x^2 + 2x - 5x - 2x^2 + 2
\]
Combine like terms:
\[
(3x^2 - 2x^2) + (2x - 5x) + 2
\]
This results in:
\[
x^2 - 3x + 2
\]
### Step 2: Factor the Numerator
Now, we will factor \( x^2 - 3x + 2 \):
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]
### Step 3: Simplify the Denominator
Now, let's look at the denominator:
\[
(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2) = (x - 2)^2(x - 1)(x + 2)^2(x + 1)
\]
### Step 4: Rewrite the Complete Expression
Now substituting in the factored numerator gives:
\[
\frac{(x - 1)(x - 2)}{(x - 2)^2(x - 1)(x + 2)^2(x + 1)}
\]
### Step 5: Cancel Common Factors
In this expression, we can cancel out \( (x - 1) \) and one \( (x - 2) \):
\[
\frac{\cancel{(x - 1)} (x - 2)}{(x - 2) \cancel{(x - 1)} (x + 2)^2 (x + 1)} = \frac{1}{(x - 2)(x + 2)^2(x + 1)}
\]
### Final Result:
Thus, the expression that is equivalent to the given one is:
\[
\boxed{\frac{1}{(x - 2)(x + 2)^2(x
make this like a 7th grader wrote it
Answers
GPT-4o mini
To simplify the expression \( \frac{3x^2 + 2x - 5x - 2x^2 + 2}{(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2)} \), let's break it down step by step, just like you'd do in seventh grade math!
### Step 1: Simplify the Numerator
First, let's focus on the top part, which is called the numerator:
\[
3x^2 + 2x - 5x - 2x^2 + 2
\]
We can combine like terms here. Group the \(x^2\) terms together and the \(x\) terms together:
\[
(3x^2 - 2x^2) + (2x - 5x) + 2
\]
Now, let's do the calculations:
- For \(3x^2 - 2x^2\), we get \(x^2\).
- For \(2x - 5x\), we get \(-3x\).
Putting it all together, we have:
\[
x^2 - 3x + 2
\]
### Step 2: Factor the Numerator
Next, we need to factor \(x^2 - 3x + 2\). We are looking for two numbers that multiply to \(2\) (the last number) and add up to \(-3\) (the middle number). Those numbers are \(-1\) and \(-2\). So, we can write:
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]
### Step 3: Simplify the Denominator
Now, let's look at the bottom part, or the denominator:
\[
(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2)
\]
Notice we can group the factors:
- We have two \( (x - 2) \) factors: \( (x - 2)(x - 2) = (x - 2)^2 \)
- We have two \( (x + 2) \) factors: \( (x + 2)(x + 2) = (x + 2)^2 \)
So, the denominator can be rewritten as:
\[
(x - 2)^2 (x - 1)(x + 2)^2 (x + 1)
\]
### Step 4: Rewrite the Whole Expression
Now we can rewrite the whole expression using what we found:
\[
\frac{(x - 1)(x - 2)}{(x - 2)^2 (x - 1)(x + 2)^2 (x + 1)}
\]
### Step 5: Cancel Out Common Factors
In this fraction, we can cancel out what’s the same on the top and bottom:
1. The \( (x - 1) \) in the numerator and denominator cancels.
2. One \( (x - 2) \) in the numerator cancels with one \( (x - 2) \) in the denominator.
After canceling, we are left with:
\[
\frac{1}{(x - 2)(x + 2)^2(x + 1)}
\]
### Final Answer
So, the final answer to our original expression is:
\[
\boxed{\frac{1}{(x - 2)(x + 2)^2(x + 1)}}
\]
Yay! We simplified it!
### Step 1: Simplify the Numerator
First, let's focus on the top part, which is called the numerator:
\[
3x^2 + 2x - 5x - 2x^2 + 2
\]
We can combine like terms here. Group the \(x^2\) terms together and the \(x\) terms together:
\[
(3x^2 - 2x^2) + (2x - 5x) + 2
\]
Now, let's do the calculations:
- For \(3x^2 - 2x^2\), we get \(x^2\).
- For \(2x - 5x\), we get \(-3x\).
Putting it all together, we have:
\[
x^2 - 3x + 2
\]
### Step 2: Factor the Numerator
Next, we need to factor \(x^2 - 3x + 2\). We are looking for two numbers that multiply to \(2\) (the last number) and add up to \(-3\) (the middle number). Those numbers are \(-1\) and \(-2\). So, we can write:
\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]
### Step 3: Simplify the Denominator
Now, let's look at the bottom part, or the denominator:
\[
(x - 2)(x - 1)(x + 2)(x - 2)(x + 1)(x + 2)
\]
Notice we can group the factors:
- We have two \( (x - 2) \) factors: \( (x - 2)(x - 2) = (x - 2)^2 \)
- We have two \( (x + 2) \) factors: \( (x + 2)(x + 2) = (x + 2)^2 \)
So, the denominator can be rewritten as:
\[
(x - 2)^2 (x - 1)(x + 2)^2 (x + 1)
\]
### Step 4: Rewrite the Whole Expression
Now we can rewrite the whole expression using what we found:
\[
\frac{(x - 1)(x - 2)}{(x - 2)^2 (x - 1)(x + 2)^2 (x + 1)}
\]
### Step 5: Cancel Out Common Factors
In this fraction, we can cancel out what’s the same on the top and bottom:
1. The \( (x - 1) \) in the numerator and denominator cancels.
2. One \( (x - 2) \) in the numerator cancels with one \( (x - 2) \) in the denominator.
After canceling, we are left with:
\[
\frac{1}{(x - 2)(x + 2)^2(x + 1)}
\]
### Final Answer
So, the final answer to our original expression is:
\[
\boxed{\frac{1}{(x - 2)(x + 2)^2(x + 1)}}
\]
Yay! We simplified it!