Question
1) Multiply and simplify the following: x+2x−1 ⋅ x−3x+2 (1 point) Responses x−1 x minus 1 1x+2 the fraction with numerator 1 and denominator x plus 2 x−3x−1 the fraction with numerator x minus 3 and denominator x minus 1 1x−1
Answers
GPT-4o mini
To multiply and simplify the expression \( (x + 2x - 1) \cdot (x - 3x + 2) \), we first simplify each expression inside the parentheses:
1. Combine like terms in the first parentheses:
\[
x + 2x - 1 = 3x - 1
\]
2. Combine like terms in the second parentheses:
\[
x - 3x + 2 = -2x + 2
\]
Now we can rewrite the product:
\[
(3x - 1)(-2x + 2)
\]
Next, we apply the distributive property (FOIL method):
\[
(3x)(-2x) + (3x)(2) + (-1)(-2x) + (-1)(2)
\]
Calculating each term:
- \( 3x \cdot -2x = -6x^2 \)
- \( 3x \cdot 2 = 6x \)
- \( -1 \cdot -2x = 2x \)
- \( -1 \cdot 2 = -2 \)
Now combine the results:
\[
-6x^2 + 6x + 2x - 2
\]
Combining the \( x \) terms:
\[
-6x^2 + 8x - 2
\]
Thus, the simplified product is:
\[
\boxed{-6x^2 + 8x - 2}
\]
1. Combine like terms in the first parentheses:
\[
x + 2x - 1 = 3x - 1
\]
2. Combine like terms in the second parentheses:
\[
x - 3x + 2 = -2x + 2
\]
Now we can rewrite the product:
\[
(3x - 1)(-2x + 2)
\]
Next, we apply the distributive property (FOIL method):
\[
(3x)(-2x) + (3x)(2) + (-1)(-2x) + (-1)(2)
\]
Calculating each term:
- \( 3x \cdot -2x = -6x^2 \)
- \( 3x \cdot 2 = 6x \)
- \( -1 \cdot -2x = 2x \)
- \( -1 \cdot 2 = -2 \)
Now combine the results:
\[
-6x^2 + 6x + 2x - 2
\]
Combining the \( x \) terms:
\[
-6x^2 + 8x - 2
\]
Thus, the simplified product is:
\[
\boxed{-6x^2 + 8x - 2}
\]
sayletyler
Answer? question is mutiply and simplify the follwoing x+2 over x-1 mutiplied by x-3 over x+2
GPT-4o mini
To multiply and simplify the expression \(\frac{x + 2}{x - 1} \cdot \frac{x - 3}{x + 2}\), follow these steps:
1. **Multiply the fractions**:
\[
\frac{x + 2}{x - 1} \cdot \frac{x - 3}{x + 2} = \frac{(x + 2)(x - 3)}{(x - 1)(x + 2)}
\]
2. **Notice that \(x + 2\) in the numerator and denominator cancels out**, assuming \(x \neq -2\):
\[
= \frac{(x - 3)}{(x - 1)}
\]
Thus, the simplified expression is:
\[
\frac{x - 3}{x - 1}
\]
So the final answer is:
\[
\boxed{\frac{x - 3}{x - 1}}
\]
Just remember that \(x\) cannot equal \(-2\) (which would make the original fraction undefined) or \(1\) (which would make the denominator in the second fraction undefined).
1. **Multiply the fractions**:
\[
\frac{x + 2}{x - 1} \cdot \frac{x - 3}{x + 2} = \frac{(x + 2)(x - 3)}{(x - 1)(x + 2)}
\]
2. **Notice that \(x + 2\) in the numerator and denominator cancels out**, assuming \(x \neq -2\):
\[
= \frac{(x - 3)}{(x - 1)}
\]
Thus, the simplified expression is:
\[
\frac{x - 3}{x - 1}
\]
So the final answer is:
\[
\boxed{\frac{x - 3}{x - 1}}
\]
Just remember that \(x\) cannot equal \(-2\) (which would make the original fraction undefined) or \(1\) (which would make the denominator in the second fraction undefined).
sayletyler
Divide and simplify x^2-4 over x+1 divided by x-2 over x+1
GPT-4o mini
To divide and simplify the expression \(\frac{x^2 - 4}{x + 1} \div \frac{x - 2}{x + 1}\), we can follow these steps:
1. **Rewrite the division as multiplication**:
Division by a fraction is equivalent to multiplying by its reciprocal:
\[
\frac{x^2 - 4}{x + 1} \div \frac{x - 2}{x + 1} = \frac{x^2 - 4}{x + 1} \cdot \frac{x + 1}{x - 2}
\]
2. **Notice that \(x + 1\) cancels out**:
As long as \(x \neq -1\):
\[
= \frac{x^2 - 4}{x - 2}
\]
The expression \(x^2 - 4\) can be factored as a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]
3. **Substitute the factored form**:
\[
= \frac{(x - 2)(x + 2)}{x - 2}
\]
4. **Cancel \(x - 2\)**:
As long as \(x \neq 2\):
\[
= x + 2
\]
So, the simplified expression is:
\[
\boxed{x + 2}
\]
Note that \(x\) cannot be \(2\) or \(-1\) to avoid undefined expressions in the original problem.
1. **Rewrite the division as multiplication**:
Division by a fraction is equivalent to multiplying by its reciprocal:
\[
\frac{x^2 - 4}{x + 1} \div \frac{x - 2}{x + 1} = \frac{x^2 - 4}{x + 1} \cdot \frac{x + 1}{x - 2}
\]
2. **Notice that \(x + 1\) cancels out**:
As long as \(x \neq -1\):
\[
= \frac{x^2 - 4}{x - 2}
\]
The expression \(x^2 - 4\) can be factored as a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]
3. **Substitute the factored form**:
\[
= \frac{(x - 2)(x + 2)}{x - 2}
\]
4. **Cancel \(x - 2\)**:
As long as \(x \neq 2\):
\[
= x + 2
\]
So, the simplified expression is:
\[
\boxed{x + 2}
\]
Note that \(x\) cannot be \(2\) or \(-1\) to avoid undefined expressions in the original problem.
sayletyler
Add and simplify x+1 over x^2-1 plus x-1 over x^2-1
GPT-4o mini
To add and simplify the expression \(\frac{x + 1}{x^2 - 1} + \frac{x - 1}{x^2 - 1}\), we can follow these steps:
1. **Identify a common denominator**:
Since both fractions have the same denominator, we can add them directly:
\[
\frac{x + 1 + (x - 1)}{x^2 - 1}
\]
2. **Combine the numerators**:
\[
= \frac{x + 1 + x - 1}{x^2 - 1}
\]
Simplifying the numerator:
\[
= \frac{2x}{x^2 - 1}
\]
3. **Factor the denominator**:
The denominator \(x^2 - 1\) is a difference of squares, which can be factored as:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
4. **Rewrite the expression**:
So now we have:
\[
= \frac{2x}{(x - 1)(x + 1)}
\]
This expression is already simplified, as \(2x\) does not have any common factors with the expression in the denominator.
### Final Answer
The final simplified expression is:
\[
\boxed{\frac{2x}{(x - 1)(x + 1)}}
\]
Note that \(x\) cannot equal \(1\) or \(-1\) in order to avoid division by zero in the original expression.
1. **Identify a common denominator**:
Since both fractions have the same denominator, we can add them directly:
\[
\frac{x + 1 + (x - 1)}{x^2 - 1}
\]
2. **Combine the numerators**:
\[
= \frac{x + 1 + x - 1}{x^2 - 1}
\]
Simplifying the numerator:
\[
= \frac{2x}{x^2 - 1}
\]
3. **Factor the denominator**:
The denominator \(x^2 - 1\) is a difference of squares, which can be factored as:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
4. **Rewrite the expression**:
So now we have:
\[
= \frac{2x}{(x - 1)(x + 1)}
\]
This expression is already simplified, as \(2x\) does not have any common factors with the expression in the denominator.
### Final Answer
The final simplified expression is:
\[
\boxed{\frac{2x}{(x - 1)(x + 1)}}
\]
Note that \(x\) cannot equal \(1\) or \(-1\) in order to avoid division by zero in the original expression.
sayletyler
Um can you look back at your work I dont think its right because thats not an option opitions are 2x over x-1
2x over x+1
2x over x^2-1
2x+2 over x-1
2x over x+1
2x over x^2-1
2x+2 over x-1
GPT-4o mini
Let’s re-evaluate the expression \(\frac{x + 1}{x^2 - 1} + \frac{x - 1}{x^2 - 1}\) to clarify the simplification:
1. **Identify the common denominator**:
Since both fractions share the same denominator (\(x^2 - 1\)), we can combine them directly:
\[
\frac{x + 1 + (x - 1)}{x^2 - 1}
\]
2. **Combine the numerators**:
\[
= \frac{(x + 1) + (x - 1)}{x^2 - 1} = \frac{x + 1 + x - 1}{x^2 - 1}
\]
\[
= \frac{2x}{x^2 - 1}
\]
3. **Factor the denominator**:
The denominator \(x^2 - 1\) can be factored using the difference of squares:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
4. **Combine the expression**:
Therefore, we have:
\[
= \frac{2x}{(x - 1)(x + 1)}
\]
Now, let's look at the provided options again:
- \(\frac{2x}{x - 1}\)
- \(\frac{2x}{x + 1}\)
- \(\frac{2x}{x^2 - 1}\)
- \(\frac{2x + 2}{x - 1}\)
The expression \(\frac{2x}{(x - 1)(x + 1)}\) can also be expressed as:
\[
\frac{2x}{x^2 - 1}
\]
This means the correct option is:
\[
\frac{2x}{x^2 - 1}
\]
### Final Answer
Thus, the correct answer is indeed:
\[
\boxed{\frac{2x}{x^2 - 1}}
\]
Thank you for your patience and for allowing me to clarify my response!
1. **Identify the common denominator**:
Since both fractions share the same denominator (\(x^2 - 1\)), we can combine them directly:
\[
\frac{x + 1 + (x - 1)}{x^2 - 1}
\]
2. **Combine the numerators**:
\[
= \frac{(x + 1) + (x - 1)}{x^2 - 1} = \frac{x + 1 + x - 1}{x^2 - 1}
\]
\[
= \frac{2x}{x^2 - 1}
\]
3. **Factor the denominator**:
The denominator \(x^2 - 1\) can be factored using the difference of squares:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
4. **Combine the expression**:
Therefore, we have:
\[
= \frac{2x}{(x - 1)(x + 1)}
\]
Now, let's look at the provided options again:
- \(\frac{2x}{x - 1}\)
- \(\frac{2x}{x + 1}\)
- \(\frac{2x}{x^2 - 1}\)
- \(\frac{2x + 2}{x - 1}\)
The expression \(\frac{2x}{(x - 1)(x + 1)}\) can also be expressed as:
\[
\frac{2x}{x^2 - 1}
\]
This means the correct option is:
\[
\frac{2x}{x^2 - 1}
\]
### Final Answer
Thus, the correct answer is indeed:
\[
\boxed{\frac{2x}{x^2 - 1}}
\]
Thank you for your patience and for allowing me to clarify my response!
sayletyler
Add and simplify 1 over x+2 plus 3 over x-2
GPT-4o mini
To add and simplify the expression \(\frac{1}{x + 2} + \frac{3}{x - 2}\), follow these steps:
1. **Find a common denominator**:
The common denominator for the two fractions is \((x + 2)(x - 2)\).
2. **Rewrite each fraction with the common denominator**:
\[
\frac{1}{x + 2} = \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)}
\]
\[
\frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)} = \frac{3x + 6}{(x - 2)(x + 2)}
\]
3. **Combine the fractions**:
Now that both fractions have the same denominator, we can add them:
\[
\frac{x - 2 + (3x + 6)}{(x + 2)(x - 2)} = \frac{x - 2 + 3x + 6}{(x + 2)(x - 2)}
\]
4. **Combine like terms in the numerator**:
\[
= \frac{(x + 3x) + (-2 + 6)}{(x + 2)(x - 2)} = \frac{4x + 4}{(x + 2)(x - 2)}
\]
5. **Factor the numerator**:
Notice that the numerator can be factored:
\[
= \frac{4(x + 1)}{(x + 2)(x - 2)}
\]
This cannot be simplified further since \(4\) and the expression in the denominator do not share any common factors.
### Final Answer
Thus, the final simplified expression is:
\[
\boxed{\frac{4(x + 1)}{(x + 2)(x - 2)}}
\]
1. **Find a common denominator**:
The common denominator for the two fractions is \((x + 2)(x - 2)\).
2. **Rewrite each fraction with the common denominator**:
\[
\frac{1}{x + 2} = \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)}
\]
\[
\frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)} = \frac{3x + 6}{(x - 2)(x + 2)}
\]
3. **Combine the fractions**:
Now that both fractions have the same denominator, we can add them:
\[
\frac{x - 2 + (3x + 6)}{(x + 2)(x - 2)} = \frac{x - 2 + 3x + 6}{(x + 2)(x - 2)}
\]
4. **Combine like terms in the numerator**:
\[
= \frac{(x + 3x) + (-2 + 6)}{(x + 2)(x - 2)} = \frac{4x + 4}{(x + 2)(x - 2)}
\]
5. **Factor the numerator**:
Notice that the numerator can be factored:
\[
= \frac{4(x + 1)}{(x + 2)(x - 2)}
\]
This cannot be simplified further since \(4\) and the expression in the denominator do not share any common factors.
### Final Answer
Thus, the final simplified expression is:
\[
\boxed{\frac{4(x + 1)}{(x + 2)(x - 2)}}
\]
sayletyler
answers are 4x+4 over x^2-4
x-2 over x+2
x+2 over 4x
4x over x+2
x-2 over x+2
x+2 over 4x
4x over x+2
GPT-4o mini
Let's go through the addition and simplification process again for the expression \( \frac{1}{x + 2} + \frac{3}{x - 2} \) to verify the previous work with the given answer options.
1. **Find a common denominator**:
The common denominator for the two fractions is \( (x + 2)(x - 2) \).
2. **Rewrite each fraction with the common denominator**:
\[
\frac{1}{x + 2} = \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)}
\]
\[
\frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)} = \frac{3x + 6}{(x - 2)(x + 2)}
\]
3. **Combine the fractions**:
Now we can add the two fractions:
\[
\frac{x - 2 + (3x + 6)}{(x + 2)(x - 2)} = \frac{x - 2 + 3x + 6}{(x + 2)(x - 2)}
\]
4. **Combine like terms in the numerator**:
\[
= \frac{(x + 3x) + (-2 + 6)}{(x + 2)(x - 2)} = \frac{4x + 4}{(x + 2)(x - 2)}
\]
5. **Recognize that the denominator \( (x + 2)(x - 2) \) can be expanded**:
\[
= \frac{4x + 4}{x^2 - 4}
\]
(since \( x^2 - 4 \) represents the product of \( (x + 2)(x - 2) \))
Now, the simplified result can indeed be expressed as:
\[
\frac{4x + 4}{x^2 - 4}
\]
### Final Answer
Therefore, the correct option is:
\[
\boxed{\frac{4x + 4}{x^2 - 4}}
\]
Thank you for your patience, and I apologize for any confusion in the previous explanations.
1. **Find a common denominator**:
The common denominator for the two fractions is \( (x + 2)(x - 2) \).
2. **Rewrite each fraction with the common denominator**:
\[
\frac{1}{x + 2} = \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)}
\]
\[
\frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)} = \frac{3x + 6}{(x - 2)(x + 2)}
\]
3. **Combine the fractions**:
Now we can add the two fractions:
\[
\frac{x - 2 + (3x + 6)}{(x + 2)(x - 2)} = \frac{x - 2 + 3x + 6}{(x + 2)(x - 2)}
\]
4. **Combine like terms in the numerator**:
\[
= \frac{(x + 3x) + (-2 + 6)}{(x + 2)(x - 2)} = \frac{4x + 4}{(x + 2)(x - 2)}
\]
5. **Recognize that the denominator \( (x + 2)(x - 2) \) can be expanded**:
\[
= \frac{4x + 4}{x^2 - 4}
\]
(since \( x^2 - 4 \) represents the product of \( (x + 2)(x - 2) \))
Now, the simplified result can indeed be expressed as:
\[
\frac{4x + 4}{x^2 - 4}
\]
### Final Answer
Therefore, the correct option is:
\[
\boxed{\frac{4x + 4}{x^2 - 4}}
\]
Thank you for your patience, and I apologize for any confusion in the previous explanations.