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Answer :
Let's carefully identify the mistake Morgan made when subtracting the rational expressions.
The original problem is:
[tex]\[
\frac{3t^2 - 4t + 1}{t + 3} - \frac{t^2 + 2t + 2}{t + 3}
\][/tex]
To subtract these rational expressions, we keep the common denominator, [tex]\(t + 3\)[/tex], and subtract the numerators:
1. Write the expression as a single fraction:
[tex]\[
\frac{(3t^2 - 4t + 1) - (t^2 + 2t + 2)}{t + 3}
\][/tex]
2. Distribute the negative sign to the entire second numerator:
[tex]\[
3t^2 - 4t + 1 - t^2 - 2t - 2
\][/tex]
3. Combine like terms:
[tex]\[
(3t^2 - t^2) + (-4t - 2t) + (1 - 2)
\][/tex]
- Combine [tex]\(3t^2 - t^2\)[/tex] to get [tex]\(2t^2\)[/tex].
- Combine [tex]\(-4t - 2t\)[/tex] to get [tex]\(-6t\)[/tex].
- Combine [tex]\(1 - 2\)[/tex] to get [tex]\(-1\)[/tex].
Therefore, the numerator becomes:
[tex]\[
2t^2 - 6t - 1
\][/tex]
4. Write the resulting expression:
[tex]\[
\frac{2t^2 - 6t - 1}{t + 3}
\][/tex]
Now, compare with Morgan's incorrect result, which is:
[tex]\[
\frac{2t^2 - 2t + 3}{t + 3}
\][/tex]
The correct numerator should be [tex]\(2t^2 - 6t - 1\)[/tex], not [tex]\(2t^2 - 2t + 3\)[/tex].
The error Morgan made was not properly distributing the negative sign to all terms in the second expression. This resulted in incorrect combining of like terms in the numerators.
Thus, the correct answer is:
- Morgan forgot to distribute the negative sign to the terms in the second expression.
The original problem is:
[tex]\[
\frac{3t^2 - 4t + 1}{t + 3} - \frac{t^2 + 2t + 2}{t + 3}
\][/tex]
To subtract these rational expressions, we keep the common denominator, [tex]\(t + 3\)[/tex], and subtract the numerators:
1. Write the expression as a single fraction:
[tex]\[
\frac{(3t^2 - 4t + 1) - (t^2 + 2t + 2)}{t + 3}
\][/tex]
2. Distribute the negative sign to the entire second numerator:
[tex]\[
3t^2 - 4t + 1 - t^2 - 2t - 2
\][/tex]
3. Combine like terms:
[tex]\[
(3t^2 - t^2) + (-4t - 2t) + (1 - 2)
\][/tex]
- Combine [tex]\(3t^2 - t^2\)[/tex] to get [tex]\(2t^2\)[/tex].
- Combine [tex]\(-4t - 2t\)[/tex] to get [tex]\(-6t\)[/tex].
- Combine [tex]\(1 - 2\)[/tex] to get [tex]\(-1\)[/tex].
Therefore, the numerator becomes:
[tex]\[
2t^2 - 6t - 1
\][/tex]
4. Write the resulting expression:
[tex]\[
\frac{2t^2 - 6t - 1}{t + 3}
\][/tex]
Now, compare with Morgan's incorrect result, which is:
[tex]\[
\frac{2t^2 - 2t + 3}{t + 3}
\][/tex]
The correct numerator should be [tex]\(2t^2 - 6t - 1\)[/tex], not [tex]\(2t^2 - 2t + 3\)[/tex].
The error Morgan made was not properly distributing the negative sign to all terms in the second expression. This resulted in incorrect combining of like terms in the numerators.
Thus, the correct answer is:
- Morgan forgot to distribute the negative sign to the terms in the second expression.
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