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Morgan made a mistake when subtracting the rational expressions below:

\[
\frac{3t^2 - 4t + 1}{t + 3} - \frac{t^2 + 2t + 2}{t + 3} = \frac{2t^2 - 2t + 3}{t + 3}
\]

What was Morgan's error?

A. Morgan forgot to combine only like terms.

B. Morgan forgot to subtract the denominators as well as numerators.

C. Morgan forgot to cancel out the +3 in the numerator and denominator as her final step.

D. Morgan forgot to distribute the negative sign to two of the terms in the second expression.

Answer :

Answer: Choice D.

Morgan forgot to distribute the negative sign to two of the terms in the second expression.

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Explanation:

Focus on the numerators.

We have (3t^2-4t+1) as the first numerator and we subtract off (t^2+2t+2) as the second numerator.

Morgan needs to simplify (3t^2-4t+1)-(t^2+2t+2) for the numerator.

Mistakenly, she had these steps

(3t^2-4t+1)-(t^2+2t+2)

3t^2-4t+1-t^2+2t+2 .... her mistake made here

(3t^2-t^2)+(-4t+2t)+(1+2)

2t^2-2t+3

All of this applies to the numerator. The denominator stays at t+3 the entire time. So effectively we can ignore it on a temporary basis.

Here's what Morgan should have for her steps when simplifying the numerator.

(3t^2-4t+1)-(t^2+2t+2)

3t^2-4t+1-t^2-2t-2 ..... distribute the negative

(3t^2-t^2)+(-4t-2t)+(1-2)

2t^2-6t-1

Note in the second step, the negative outside flips the sign of each term in the second parenthesis.

Therefore,

[tex]\frac{3t^2-4t+1}{t+3}-\frac{t^2+2t+2}{t+3}\\\\\frac{(3t^2-4t+1)-(t^2+2t+2)}{t+3}\\\\\frac{3t^2-4t+1-t^2-2t-2}{t+3}\\\\\frac{2t^2-6t-1}{t+3}\\\\[/tex]

which means [tex]\frac{3t^2-4t+1}{t+3}-\frac{t^2+2t+2}{t+3}=\frac{2t^2-6t-1}{t+3}, \ \ \text{ where } t \ne -3\\\\[/tex]

Side notes:

  • The fractions can only be subtracted since the denominators are the same.
  • We have [tex]t \ne -3[/tex] to avoid a division by zero error.
  • Rational expressions are a fraction, or ratio, of two polynomials.

Thanks for taking the time to read Morgan made a mistake when subtracting the rational expressions below frac 3t 2 4t 1 t 3 frac t 2 2t 2 t 3 frac. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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