A student attempts to rewrite the expression $\frac{5 - x}{x^2 - 25}$ by factoring. Carefully accounting for the negative sign and subtraction order convention, express $\frac{5 - x}{x^2 - 25}$ in an equivalent reduced form $\frac{c}{x + a}$, where $c$ and $a$ are real constants.

A student attempts to rewrite the expression $\frac{5 - x}{x^2 - 25}$ by factoring. Carefully accounting for the negative sign and subtraction order convention, express $\frac{5 - x}{x^2 - 25}$ in an equivalent reduced form $\frac{c}{x + a}$ , where $c$ and $a$ are real constants.

Final answer

$\frac{-1}{x + 5}$ (valid for $x \neq 5$ and $x \neq -5$ )

Steps

Factor the denominator

To simplify the rational expression $\frac{5 - x}{x^2 - 25}$ , factor both the numerator and denominator to identify and cancel any common factors.

The denominator is a difference of two squares: $x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5)$

Factor out a negative sign from the numerator

The numerator is $5 - x$ . By the subtraction order identity, $a - b = -(b - a)$ . Factoring out $-1$ gives: $5 - x = -(x - 5)$

Common Pitfall: A frequent mistake is assuming $(5 - x)$ and $(x - 5)$ are identical and canceling them to get $+1$ . Because subtraction is non-commutative ( $a - b \neq b - a$ ), their quotient is always $-1$ for any value where $x \neq 5$ : $\frac{5 - x}{x - 5} = \frac{-(x - 5)}{x - 5} = -1$

Rewrite and simplify the expression

Substitute both factored forms into the original fraction: $\frac{5 - x}{x^2 - 25} = \frac{-(x - 5)}{(x - 5)(x + 5)}$

Cancel the common factor $(x - 5)$ from the numerator and denominator: $\frac{-(x - 5)}{(x - 5)(x + 5)} = \frac{-1}{x + 5}$

State restrictions on the variable

The original expression is undefined wherever the denominator is zero: $x^2 - 25 = 0 \implies x = 5 \quad \text{or} \quad x = -5$

Thus, the simplified form is $\frac{-1}{x + 5}$ , with $c = -1$ and $a = 5$ , valid for $x \neq 5$ and $x \neq -5$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student treats subtraction as commutative and equates 5 - x directly to x - 5, losing the factor of -1.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{5 - x}{x^2 - 25}$ = $\frac{x - 5}{(x - 5)(x + 5)}$
Correct
$\frac{5 - x}{x^2 - 25}$ = $\frac{-(x - 5)}{(x - 5)(x + 5)}$
You correctly factored the denominator $x^2 - 25$ as $(x - 5)(x + 5)$ , but $5 - x$ is not equal to $x - 5$ . Reversing subtraction order requires factoring out a negative sign: $5 - x = -(x - 5)$ .
Step 2
= $\frac{1}{x + 5}$
Correct
= $\frac{-1}{x + 5}$
Your cancellation of $(x - 5)$ followed logically from your previous expression, but because the numerator was missing the $-1$ factor, the resulting fraction has the incorrect sign.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly factored the difference of squares $x^2 - 25$ into $(x - 5)(x + 5)$ .

You correctly identified and canceled the matching linear factor in numerator and denominator.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign Error / Subtraction Non-Commutativity

You treated $5 - x$ as directly identical to $x - 5$ , omitting the negative sign.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking that subtraction is non-commutative ( $a - b \neq b - a$ ), which means opposite binomial terms differ by a factor of $-1$ : $a - b = -(b - a)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Opposite Factors Rule: For any real numbers, $a - b = -(b - a)$ , so $\frac{a - b}{b - a} = -1$ when $a \neq b$ .
Whenever you see reversed binomial differences like $(k - x)$ and $(x - k)$ , factor out $-1$ explicitly before canceling.

CORRECTED PATH

ACEpath Teacher's Demonstration

Factor out $-1$ from the numerator: $5 - x = -(x - 5)$ .
Substitute into the factored fraction: $\frac{-(x - 5)}{(x - 5)(x + 5)}$ .
Cancel the common factor $(x - 5)$ , leaving $-1$ in the numerator.

Final answer

$\frac{-1}{x + 5}$

NEXT STEPS

Final Takeaways from Your Teacher

Whenever you need to reverse subtraction order, write $-(b - a)$ as an explicit intermediate step.
Do a quick mental numerical check: pick a simple number like $x = 0$ to verify signs ( $5/(-25) = -1/5$ ).

OverviewACEpath Teacher's Diagnosis

02 The student factors -1 out of the numerator but leaves the negative sign in the denominator without distributing or preserving it as -1 in the numerator.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{-(x - 5)}{(x - 5)(x + 5)}$
Step 2
$\frac{1}{x - 5}$
Correct
$\frac{-1}{x + 5}$
Canceling the common factor $(x - 5)$ from both the numerator and denominator leaves $-1$ in the numerator and $(x + 5)$ in the denominator, giving $\frac{-1}{x + 5}$ rather than $\frac{1}{x - 5}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You recognized that $5 - x$ can be rewritten as $-(x - 5)$ by factoring out $-1$ .

You correctly factored the difference of squares $x^2 - 25$ into $(x - 5)(x + 5)$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Cancellation and Sign Tracking Error

When dividing out $(x - 5)$ , you discarded the negative sign and mistakenly kept $(x - 5)$ instead of $(x + 5)$ in the denominator.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of remaining terms and factors during rational expression cancellation, particularly mistaking which factor cancels with the numerator and dropping the leading $-1$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Track the Negative Factor: When a factor $-(A)$ cancels with $A$ , the numerator retains a factor of $-1$ , not $+1$ .
Write $\frac{-1 \cdot (x - 5)}{(x - 5)(x + 5)}$ explicitly to see that $-1$ remains in the numerator.
Match Identical Factors to Cancel: Only identical factors cancel to $1$ : $\frac{x - 5}{x - 5} = 1$ .
Cross out the pair $(x - 5)$ in both parts, then clearly write the remaining factor $(x + 5)$ in the denominator.

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide numerator and denominator by the common factor $(x - 5)$ for $x \neq 5$ :
$\frac{-(x - 5)}{(x - 5)(x + 5)} = \frac{-1 \cdot 1}{1 \cdot (x + 5)} = \frac{-1}{x + 5}$

Final answer

$\frac{-1}{x + 5}$

NEXT STEPS

Final Takeaways from Your Teacher

Lightly slash through the exact identical factors you are canceling to avoid confusing $(x - 5)$ with $(x + 5)$ .
Always write the remaining numerical factor (such as $-1$ ) explicitly in the numerator before writing your final fraction.

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly factors the difference of squares x² - 25 as (5 - x)(5 + x) without accounting for the leading variable sign.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{5 - x}{x^2 - 25}$ = $\frac{5 - x}{(5 - x)(5 + x)}$
Correct
$\frac{5 - x}{(x - 5)(x + 5)}$ = $\frac{-(x - 5)}{(x - 5)(x + 5)}$
You factored the difference of squares in reverse order. Expanding your denominator gives $(5 - x)(5 + x) = 25 - x^2$ , which is the opposite of $x^2 - 25$ . Factoring $x^2 - 25$ correctly yields $(x - 5)(x + 5)$ .
Step 2
= $\frac{1}{x + 5}$
Correct
= $\frac{-1}{x + 5}$
You correctly canceled the identical factor $(5 - x)$ and commuted the addition in the denominator, but this step inherited the missing negative sign from Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly identified that the denominator is a difference of squares.

You canceled identical binomial factors between the numerator and denominator.

You correctly simplified the remaining addition in the denominator from $5 + x$ to $x + 5$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Reversed difference-of-squares terms

You factored $x^2 - 25$ as $(5 - x)(5 + x)$ instead of $(x - 5)(x + 5)$ , matching the order of the numerator instead of following the order of the original expression.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the order of terms in subtraction: treating $a^2 - b^2$ and $b^2 - a^2$ as interchangeable because addition is commutative, forgetting that $a - b = -(b - a)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Difference of Squares Order: $a^2 - b^2 = (a - b)(a + b)$ , while $b^2 - a^2 = (b - a)(b + a) = -(a^2 - b^2)$ .
Keep the first term first and the second term second when writing the factors $(a - b)(a + b)$ .
Opposite Binomial Identity: For any two terms, $b - a = -(a - b)$ , so $\frac{b - a}{a - b} = -1$ when $a \neq b$ .
When you see reversed subtraction terms, factor out $-1$ before canceling.

CORRECTED PATH

ACEpath Teacher's Demonstration

Factor the denominator in its true order: $x^2 - 25 = (x - 5)(x + 5)$ .
Rewrite the numerator by factoring out $-1$ : $5 - x = -(x - 5)$ .
Form the rational expression: $\frac{-(x - 5)}{(x - 5)(x + 5)}$ .
Cancel the common factor $(x - 5)$ .

Final answer

$\frac{-1}{x + 5}$

NEXT STEPS

Final Takeaways from Your Teacher

Double-check your factored forms by mentally expanding them before canceling any terms.
Practice identifying opposite binomial pairs like $(x - a)$ and $(a - x)$ and extracting a $-1$ factor explicitly.

Lock it in

Question

What is the algebraic relationship between $a - b$ and $b - a$ ?

Answer

$a - b = -(b - a)$ . Because subtraction is non-commutative, reversing the order changes the sign.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student treats subtraction as commutative and equates 5 - x directly to x - 5, losing the factor of -1.

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 factors -1 out of the numerator but leaves the negative sign in the denominator without distributing or preserving it as -1 in the numerator.

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 incorrectly factors the difference of squares x² - 25 as (5 - x)(5 + x) without accounting for the leading variable sign.

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

Similar question