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Answer :
To determine Morgan's error in subtracting the rational expressions, let's break down the subtraction step by step:
We have the expressions:
[tex]\[
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 2t + 2}{t+3}
\][/tex]
Since the denominators are the same, we keep the denominator and subtract the numerators:
[tex]\[
(3t^2 - 4t + 1) - (t^2 + 2t + 2)
\][/tex]
Here’s how we should handle the subtraction of the numerators:
1. Distribute the negative sign to the second numerator:
- The expression becomes:
[tex]\[
3t^2 - 4t + 1 - t^2 - 2t - 2
\][/tex]
2. Combine like terms:
- For the [tex]\( t^2 \)[/tex] terms:
[tex]\[
3t^2 - t^2 = 2t^2
\][/tex]
- For the [tex]\( t \)[/tex] terms:
[tex]\[
-4t - 2t = -6t
\][/tex]
- For the constant terms:
[tex]\[
1 - 2 = -1
\][/tex]
Thus, the correctly subtracted expression for the numerator is:
[tex]\[
2t^2 - 6t - 1
\][/tex]
The resulting rational expression then becomes:
[tex]\[
\frac{2t^2 - 6t - 1}{t+3}
\][/tex]
Morgan's original answer was:
[tex]\[
\frac{2t^2 - 2t + 3}{t+3}
\][/tex]
Comparing the numerators reveals that Morgan's mistake was in failing to correctly distribute the negative sign to the terms in the second numerator, specifically to the [tex]\( 2t \)[/tex] and the constant [tex]\( 2 \)[/tex].
Thus, the error can be clearly identified as: Morgan forgot to distribute the negative sign to two of the terms in the second expression.
We have the expressions:
[tex]\[
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 2t + 2}{t+3}
\][/tex]
Since the denominators are the same, we keep the denominator and subtract the numerators:
[tex]\[
(3t^2 - 4t + 1) - (t^2 + 2t + 2)
\][/tex]
Here’s how we should handle the subtraction of the numerators:
1. Distribute the negative sign to the second numerator:
- The expression becomes:
[tex]\[
3t^2 - 4t + 1 - t^2 - 2t - 2
\][/tex]
2. Combine like terms:
- For the [tex]\( t^2 \)[/tex] terms:
[tex]\[
3t^2 - t^2 = 2t^2
\][/tex]
- For the [tex]\( t \)[/tex] terms:
[tex]\[
-4t - 2t = -6t
\][/tex]
- For the constant terms:
[tex]\[
1 - 2 = -1
\][/tex]
Thus, the correctly subtracted expression for the numerator is:
[tex]\[
2t^2 - 6t - 1
\][/tex]
The resulting rational expression then becomes:
[tex]\[
\frac{2t^2 - 6t - 1}{t+3}
\][/tex]
Morgan's original answer was:
[tex]\[
\frac{2t^2 - 2t + 3}{t+3}
\][/tex]
Comparing the numerators reveals that Morgan's mistake was in failing to correctly distribute the negative sign to the terms in the second numerator, specifically to the [tex]\( 2t \)[/tex] and the constant [tex]\( 2 \)[/tex].
Thus, the error can be clearly identified as: Morgan forgot to distribute the negative sign to two of the terms in the second expression.
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