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Answer :
We start with the expression
[tex]$$
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 2t + 2}{t+3}.
$$[/tex]
Since the denominators are the same, the subtraction can be performed by subtracting the second numerator from the first:
[tex]$$
\frac{(3t^2 - 4t + 1) - (t^2 + 2t + 2)}{t+3}.
$$[/tex]
The critical step is to subtract each term in the second numerator. The negative sign must be distributed to every term inside the parentheses:
[tex]$$
(3t^2 - 4t + 1) - (t^2 + 2t + 2) = 3t^2 - 4t + 1 - t^2 - 2t - 2.
$$[/tex]
Next, we combine like terms:
1. For the [tex]$t^2$[/tex] terms:
[tex]$$
3t^2 - t^2 = 2t^2.
$$[/tex]
2. For the [tex]$t$[/tex] terms:
[tex]$$
-4t - 2t = -6t.
$$[/tex]
3. For the constants:
[tex]$$
1 - 2 = -1.
$$[/tex]
Thus, the simplified numerator becomes
[tex]$$
2t^2 - 6t - 1,
$$[/tex]
and the expression is
[tex]$$
\frac{2t^2 - 6t - 1}{t+3}.
$$[/tex]
Morgan's error was in the subtraction step. She did not distribute the negative sign correctly to the terms in the second numerator, which would lead to an incorrect combination of like terms and an erroneous result.
In conclusion, the mistake was that Morgan forgot to distribute the negative sign to two of the terms in the second expression.
[tex]$$
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 2t + 2}{t+3}.
$$[/tex]
Since the denominators are the same, the subtraction can be performed by subtracting the second numerator from the first:
[tex]$$
\frac{(3t^2 - 4t + 1) - (t^2 + 2t + 2)}{t+3}.
$$[/tex]
The critical step is to subtract each term in the second numerator. The negative sign must be distributed to every term inside the parentheses:
[tex]$$
(3t^2 - 4t + 1) - (t^2 + 2t + 2) = 3t^2 - 4t + 1 - t^2 - 2t - 2.
$$[/tex]
Next, we combine like terms:
1. For the [tex]$t^2$[/tex] terms:
[tex]$$
3t^2 - t^2 = 2t^2.
$$[/tex]
2. For the [tex]$t$[/tex] terms:
[tex]$$
-4t - 2t = -6t.
$$[/tex]
3. For the constants:
[tex]$$
1 - 2 = -1.
$$[/tex]
Thus, the simplified numerator becomes
[tex]$$
2t^2 - 6t - 1,
$$[/tex]
and the expression is
[tex]$$
\frac{2t^2 - 6t - 1}{t+3}.
$$[/tex]
Morgan's error was in the subtraction step. She did not distribute the negative sign correctly to the terms in the second numerator, which would lead to an incorrect combination of like terms and an erroneous result.
In conclusion, the mistake was that Morgan forgot to distribute the negative sign to two of the terms in the second expression.
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