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Answer :
* Distribute the negative sign: $(3t^2 - 4t + 1) - (t^2 + 2t + 2) = 3t^2 - 4t + 1 - t^2 - 2t - 2$.
* Combine like terms: $(3t^2 - t^2) + (-4t - 2t) + (1 - 2)$.
* Simplify the expression: $2t^2 - 6t - 1$.
* Morgan forgot to distribute the negative sign to two of the terms in the second expression, so the answer is $\boxed{\text{Morgan forgot to distribute the negative sign to two of the terms in the second expression.}}$
### Explanation
1. Problem Analysis
We are given the subtraction problem $\frac{3 t^2-4 t+1}{t+3}-\frac{t^2+2 t+2}{t+3}=\frac{2 t^2-2 t+3}{t+3}$ and asked to identify Morgan's mistake. Since the denominators are the same, we can subtract the numerators.
2. Distributing the Negative Sign
The correct subtraction of the numerators is $(3t^2 - 4t + 1) - (t^2 + 2t + 2)$. We need to distribute the negative sign to each term in the second expression.
3. Applying the Distribution
Distributing the negative sign, we get $3t^2 - 4t + 1 - t^2 - 2t - 2$.
4. Combining Like Terms
Now, we combine like terms: $(3t^2 - t^2) + (-4t - 2t) + (1 - 2)$.
5. Correct Result
Simplifying, we have $2t^2 - 6t - 1$. Therefore, the correct result of the subtraction is $\frac{2t^2 - 6t - 1}{t+3}$.
6. Identifying the Error
Morgan's result was $\frac{2 t^2-2 t+3}{t+3}$. Comparing this to the correct result $\frac{2t^2 - 6t - 1}{t+3}$, we see that Morgan made a mistake in combining the terms. Specifically, she did not correctly distribute the negative sign to all terms in the second expression.
7. Conclusion
Let's examine the given options:
* Morgan forgot to combine only like terms.
* Morgan forgot to subtract the denominators as well as the numerators.
* Morgan forgot to cancel out the +3 in the numerator and denominator as her final step.
* Morgan forgot to distribute the negative sign to two of the terms in the second expression.
The correct answer is that Morgan forgot to distribute the negative sign to two of the terms in the second expression.
### Examples
When simplifying complex recipes, it's crucial to accurately subtract ingredients. Just like Morgan's mistake in subtracting rational expressions, forgetting to distribute the negative sign in a recipe could lead to an incorrect amount of ingredients, resulting in a dish that doesn't taste as expected. Attention to detail in mathematical operations, like distributing negative signs, ensures accuracy in various real-life applications, from cooking to engineering.
* Combine like terms: $(3t^2 - t^2) + (-4t - 2t) + (1 - 2)$.
* Simplify the expression: $2t^2 - 6t - 1$.
* Morgan forgot to distribute the negative sign to two of the terms in the second expression, so the answer is $\boxed{\text{Morgan forgot to distribute the negative sign to two of the terms in the second expression.}}$
### Explanation
1. Problem Analysis
We are given the subtraction problem $\frac{3 t^2-4 t+1}{t+3}-\frac{t^2+2 t+2}{t+3}=\frac{2 t^2-2 t+3}{t+3}$ and asked to identify Morgan's mistake. Since the denominators are the same, we can subtract the numerators.
2. Distributing the Negative Sign
The correct subtraction of the numerators is $(3t^2 - 4t + 1) - (t^2 + 2t + 2)$. We need to distribute the negative sign to each term in the second expression.
3. Applying the Distribution
Distributing the negative sign, we get $3t^2 - 4t + 1 - t^2 - 2t - 2$.
4. Combining Like Terms
Now, we combine like terms: $(3t^2 - t^2) + (-4t - 2t) + (1 - 2)$.
5. Correct Result
Simplifying, we have $2t^2 - 6t - 1$. Therefore, the correct result of the subtraction is $\frac{2t^2 - 6t - 1}{t+3}$.
6. Identifying the Error
Morgan's result was $\frac{2 t^2-2 t+3}{t+3}$. Comparing this to the correct result $\frac{2t^2 - 6t - 1}{t+3}$, we see that Morgan made a mistake in combining the terms. Specifically, she did not correctly distribute the negative sign to all terms in the second expression.
7. Conclusion
Let's examine the given options:
* Morgan forgot to combine only like terms.
* Morgan forgot to subtract the denominators as well as the numerators.
* Morgan forgot to cancel out the +3 in the numerator and denominator as her final step.
* Morgan forgot to distribute the negative sign to two of the terms in the second expression.
The correct answer is that Morgan forgot to distribute the negative sign to two of the terms in the second expression.
### Examples
When simplifying complex recipes, it's crucial to accurately subtract ingredients. Just like Morgan's mistake in subtracting rational expressions, forgetting to distribute the negative sign in a recipe could lead to an incorrect amount of ingredients, resulting in a dish that doesn't taste as expected. Attention to detail in mathematical operations, like distributing negative signs, ensures accuracy in various real-life applications, from cooking to engineering.
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