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Answer :
To solve the question correctly, let's look at the steps needed to subtract two rational expressions.
When subtracting rational expressions that have the same denominator, you only need to focus on the numerators.
The given expression is:
[tex]\[
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 9t + 2}{t+3}
\][/tex]
Step-by-step Solution:
1. Combine the Numerators:
Since both expressions have the same denominator [tex]\((t + 3)\)[/tex], you can subtract the numerators directly.
2. Subtract the Numerators:
You need to subtract the entire second numerator [tex]\((t^2 + 9t + 2)\)[/tex] from the first numerator [tex]\((3t^2 - 4t + 1)\)[/tex].
This means you should distribute the negative sign to all terms in the second numerator:
[tex]\[
3t^2 - 4t + 1 - (t^2 + 9t + 2)
\][/tex]
Distribute the negative sign:
[tex]\[
3t^2 - 4t + 1 - t^2 - 9t - 2
\][/tex]
3. Combine Like Terms:
Combine the like terms from the distributed expression:
- Combine the [tex]\(t^2\)[/tex] terms: [tex]\(3t^2 - t^2 = 2t^2\)[/tex]
- Combine the [tex]\(t\)[/tex] terms: [tex]\(-4t - 9t = -13t\)[/tex]
- Combine the constant terms: [tex]\(1 - 2 = -1\)[/tex]
The new numerator becomes:
[tex]\[
2t^2 - 13t - 1
\][/tex]
4. Rewrite the Expression:
The correct subtraction of the rational expressions is:
[tex]\[
\frac{2t^2 - 13t - 1}{t+3}
\][/tex]
Conclusion:
Morgan's error was indeed a failure to distribute the negative sign correctly to the second expression's terms.
When subtracting rational expressions that have the same denominator, you only need to focus on the numerators.
The given expression is:
[tex]\[
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 9t + 2}{t+3}
\][/tex]
Step-by-step Solution:
1. Combine the Numerators:
Since both expressions have the same denominator [tex]\((t + 3)\)[/tex], you can subtract the numerators directly.
2. Subtract the Numerators:
You need to subtract the entire second numerator [tex]\((t^2 + 9t + 2)\)[/tex] from the first numerator [tex]\((3t^2 - 4t + 1)\)[/tex].
This means you should distribute the negative sign to all terms in the second numerator:
[tex]\[
3t^2 - 4t + 1 - (t^2 + 9t + 2)
\][/tex]
Distribute the negative sign:
[tex]\[
3t^2 - 4t + 1 - t^2 - 9t - 2
\][/tex]
3. Combine Like Terms:
Combine the like terms from the distributed expression:
- Combine the [tex]\(t^2\)[/tex] terms: [tex]\(3t^2 - t^2 = 2t^2\)[/tex]
- Combine the [tex]\(t\)[/tex] terms: [tex]\(-4t - 9t = -13t\)[/tex]
- Combine the constant terms: [tex]\(1 - 2 = -1\)[/tex]
The new numerator becomes:
[tex]\[
2t^2 - 13t - 1
\][/tex]
4. Rewrite the Expression:
The correct subtraction of the rational expressions is:
[tex]\[
\frac{2t^2 - 13t - 1}{t+3}
\][/tex]
Conclusion:
Morgan's error was indeed a failure to distribute the negative sign correctly to the second expression's terms.
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