A rational function is known to decompose into partial fractions of the form $\frac{A}{x - 3} + \frac{B}{x + 4}$. When evaluated, the decomposition yields $\frac{2}{x - 3} - \frac{5}{x + 4}$. Reconstruct the original expression by combining these partial fractions into a single equivalent rational expression $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are expanded polynomials.

A rational function is known to decompose into partial fractions of the form $\frac{A}{x - 3} + \frac{B}{x + 4}$ . When evaluated, the decomposition yields $\frac{2}{x - 3} - \frac{5}{x + 4}$ . Reconstruct the original expression by combining these partial fractions into a single equivalent rational expression $\frac{P(x)}{Q(x)}$ , where $P(x)$ and $Q(x)$ are expanded polynomials.

Final answer

$\frac{-3x + 23}{x^2 + x - 12}$

Steps

Find the common denominator

To recombine partial fractions into a single rational expression, follow the process of adding or subtracting fractions by finding a common denominator.

The denominators are $(x - 3)$ and $(x + 4)$ . Since these two linear factors share no common factors, the least common denominator (LCD) is their product: $\text{LCD} = (x - 3)(x + 4)$

Rewrite each fraction with the common denominator

Multiply the numerator and denominator of each fraction by the factor missing from its denominator: $\frac{2}{x - 3} - \frac{5}{x + 4} = \frac{2(x + 4)}{(x - 3)(x + 4)} - \frac{5(x - 3)}{(x - 3)(x + 4)}$

Combine and expand the numerators

Combine the numerators over the single common denominator: $\frac{2(x + 4) - 5(x - 3)}{(x - 3)(x + 4)}$

Expand the terms in the numerator, being careful to distribute the negative sign across both terms of $-5(x - 3)$ : $2(x + 4) - 5(x - 3) = 2x + 8 - 5x + 15$

Group like terms: $(2x - 5x) + (8 + 15) = -3x + 23$

Common Pitfall: A frequent mistake when subtracting rational expressions is failing to distribute the negative sign to all terms in the second numerator. Make sure $-5(x - 3)$ becomes $-5x + 15$ , not $-5x - 15$ .

Expand the denominator

Expand the denominator into standard polynomial form: $(x - 3)(x + 4) = x^2 + 4x - 3x - 12 = x^2 + x - 12$

Conclusion

Putting the expanded numerator and denominator together yields the single rational expression: $\frac{-3x + 23}{x^2 + x - 12}$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student fails to distribute the negative sign to the second term when expanding -5(x - 3), writing -15 instead of +15.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{2}{x - 3}$ - $\frac{5}{x + 4}$ = $\frac{2(x + 4) - 5(x - 3)}{(x - 3)(x + 4)}$
Step 2
$\frac{2(x + 4) - 5(x - 3)}{(x - 3)(x + 4)}$ = $\frac{2x + 8 - 5x - 15}{x^2 + x - 12}$
Correct
$\frac{2x + 8 - 5x + 15}{x^2 + x - 12}$
When expanding $-5(x - 3)$ , multiplying the negative factor $-5$ by $-3$ yields $+15$ , but you wrote $-15$ .
Step 3
$\frac{2x + 8 - 5x - 15}{x^2 + x - 12}$ = $\frac{-3x - 7}{x^2 + x - 12}$
Correct
$\frac{-3x + 23}{x^2 + x - 12}$
You simplified $8 - 15$ to $-7$ correctly based on your previous line, but the result is affected by the earlier sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You accurately identified the common denominator as $(x - 3)(x + 4)$ .

You correctly multiplied out the denominator to obtain $x^2 + x - 12$ .

You combined like terms consistently with your written expressions.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error

You wrote $-15$ instead of $+15$ when multiplying $-5$ by $-3$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Distributing a negative factor across subtraction requires applying the negative sign to both terms: $-a(b - c) = -ab + ac$ . Overlooking the double negative led to an incorrect sign on the constant term.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute the sign with the coefficient: Treat subtraction preceding a grouped term as multiplying by a negative number: $-(a - b) = -a + b$ .
Circle the minus sign together with the factor $-5$ , and explicitly multiply: $(-5) \cdot (x) = -5x$ and $(-5) \cdot (-3) = +15$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the numerator correctly: $2(x + 4) - 5(x - 3) = 2x + 8 - 5x + 15$ .
Combine like terms in the numerator: $(2x - 5x) + (8 + 15) = -3x + 23$ .
Place the expanded numerator over the expanded denominator: $\frac{-3x + 23}{x^2 + x - 12}$ .

Final answer

$\frac{-3x + 23}{x^2 + x - 12}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative numbers across parentheses with subtraction, such as $-4(x - 2)$ .
When subtracting rational expressions, rewrite $-\frac{A}{B}$ as $+ \frac{-A}{B}$ before expanding to prevent sign slips.

OverviewACEpath Teacher's Diagnosis

02 The student adds the numerators and denominators directly instead of finding a common denominator.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{2}{x-3} - \frac{5}{x+4} = \frac{2-5}{(x-3)+(x+4)}$
Correct
$\frac{2(x+4)}{(x-3)(x+4)}$ - $\frac{5(x-3)}{(x-3)(x+4)}$ = $\frac{2(x+4) - 5(x-3)}{(x-3)(x+4)}$
Fractions cannot be combined by subtracting numerators and adding denominators. To subtract rational expressions, you must first find a common denominator, which is $(x - 3)(x + 4)$ , and scale each numerator accordingly.
Step 2
$= \frac{-3}{2x+1}$
Correct
= $\frac{-3x + 23}{x^2 + x - 12}$
Your arithmetic $2 - 5 = -3$ and $(x - 3) + (x + 4) = 2x + 1$ was carried out correctly on the previous line, but because the setup in Step 1 was invalid, this final expression is incorrect.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly transcribed the initial subtraction of the two partial fractions.

You accurately simplified the arithmetic operations $2 - 5 = -3$ and $(x - 3) + (x + 4) = 2x + 1$ within your chosen expression.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Combining Fractions Without a Common Denominator

You combined the two fractions by subtracting the numerators and adding the denominators directly.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Fractions represent parts of a whole defined by their denominator, so addition and subtraction can only occur when the denominators are identical. Operating across numerators and denominators independently ( $\frac{a}{b} - \frac{c}{d} = \frac{a - c}{b + d}$ ) violates the fundamental definition of fraction arithmetic.

REASONING PATH

Your ACEpath teacher’s suggested approach

Identify the Least Common Denominator (LCD): For distinct linear denominators $(x - a)$ and $(x - b)$ , $\text{LCD} = (x - a)(x - b)$ .
Always find the product of distinct irreducible denominator factors before attempting to combine numerators.
Scale Each Numerator: $\frac{A}{B}$ $\pm$ $\frac{C}{D}$ = $\frac{A \cdot D \pm C \cdot B}{B \cdot D}$
Multiply the numerator of each fraction by the factor missing from its denominator.

CORRECTED PATH

ACEpath Teacher's Demonstration

Write the fractions with the common denominator $(x - 3)(x + 4)$ : $\frac{2(x + 4)}{(x - 3)(x + 4)} - \frac{5(x - 3)}{(x - 3)(x + 4)}$
Combine over the single denominator and expand the numerators: $\frac{2(x + 4) - 5(x - 3)}{(x - 3)(x + 4)} = \frac{2x + 8 - 5x + 15}{x^2 + 4x - 3x - 12}$
Combine like terms in both the numerator and denominator: $\frac{-3x + 23}{x^2 + x - 12}$

Final answer

$\frac{-3x + 23}{x^2 + x - 12}$

NEXT STEPS

Final Takeaways from Your Teacher

Review fraction arithmetic with simple numbers (for example, notice that $\frac{1}{2} + \frac{1}{3} \neq \frac{2}{5}$ ).
Practice multiplying each term by a 'smart form of 1' (such as $\frac{x+4}{x+4}$ ) to build common denominators systematically.
Remember to distribute negative signs across entire grouped expressions when subtracting numerators.

OverviewACEpath Teacher's Diagnosis

03 The student cross-multiplies terms in the numerator in the reverse order, subtracting the first product from the second.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{5(x - 3) - 2(x + 4)}{(x - 3)(x + 4)}$
Correct
$\frac{2(x + 4) - 5(x - 3)}{(x - 3)(x + 4)}$
When subtracting rational expressions of the form $\frac{A}{B} - \frac{C}{D}$ , the numerator must be formed as $A \cdot D - C \cdot B$ . You wrote the second cross-product minus the first ( $C \cdot B - A \cdot D$ ), which reverses the subtraction and negates the entire numerator.
Step 2
$\frac{5x - 15 - 2x - 8}{x^2 + x - 12}$
Correct
$\frac{2x + 8 - 5x + 15}{x^2 + x - 12}$
Your algebraic distribution and denominator expansion were completely correct based on your previous numerator, but this step is affected by the reversed terms in Step 1.
Step 3
$\frac{3x - 23}{x^2 + x - 12}$
Correct
$\frac{-3x + 23}{x^2 + x - 12}$
You collected like terms accurately, but the resulting polynomial has opposite signs in the numerator due to the initial subtraction order error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly identified $(x - 3)(x + 4)$ as the common denominator.

You expanded the binomial product in the denominator correctly to $x^2 + x - 12$ .

You correctly distributed signs and combined like terms across your linear expressions.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Reversed Subtraction Order in Numerator

You swapped the order of subtraction in the numerator, subtracting the first product from the second instead of the second from the first.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Subtraction is non-commutative ( $A - B \neq B - A$ ). When combining fractions using cross-multiplication or common denominators, the first numerator must lead: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$ . Swapping them produces $-(ad - bc)$ , which changes the sign of every term in the numerator.

REASONING PATH

Your ACEpath teacher’s suggested approach

Lead with the First Fraction: For any subtraction $\frac{A}{B} - \frac{C}{D}$ , the numerator is always $(\text{first numerator} \times \text{second denominator}) - (\text{second numerator} \times \text{first denominator})$ .
Always write the numerator of the left fraction first: $A(D) - C(B)$ .
Sign Check via Substitution: Evaluating the original expression and the combined expression at a simple value of $x$ should yield identical numbers.
Test $x = 0$ : the original gives $\frac{2}{-3} - \frac{5}{4} = -\frac{23}{12}$ , whereas your numerator gave $\frac{-23}{-12} = \frac{23}{12}$ , immediately exposing a sign flip.

CORRECTED PATH

ACEpath Teacher's Demonstration

Set up the combined numerator preserving the original order: $\frac{2(x + 4) - 5(x - 3)}{(x - 3)(x + 4)}$ .
Distribute both coefficients into their respective parentheses: $\frac{2x + 8 - 5x + 15}{(x - 3)(x + 4)}$ .
Combine like terms in the numerator and expand the denominator: $\frac{-3x + 23}{x^2 + x - 12}$ .

Final answer

$\frac{-3x + 23}{x^2 + x - 12}$

NEXT STEPS

Final Takeaways from Your Teacher

Write out the two fractions with their explicit scaling factors before combining into a single fraction line: $\frac{2(x+4)}{\text{LCD}} - \frac{5(x-3)}{\text{LCD}}$ .
Double-check that the subtraction sign stays attached to the second fraction's terms.
Practice combining fractions with opposite signs to reinforce the order of operations.

Lock it in

Question

What is the result of expanding $-c(x - d)$ ?

Answer

$-cx + cd$ . Multiplying two negative quantities $(-c) \cdot (-d)$ produces a positive product.

Resources

Answer and steps

Final answer

Steps

Conclusion

Typical errors

01 The student fails to distribute the negative sign to the second term when expanding -5(x - 3), writing -15 instead of +15.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student adds the numerators and denominators directly instead of finding a common denominator.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student cross-multiplies terms in the numerator in the reverse order, subtracting the first product from the second.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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