What are excluded values of a rational expression?

Excluded values (also called restrictions on the domain) are values of x that make the denominator equal to zero. They are found by setting Q(x)=0 and solving. For (x+1)/[(x−2)(x+3)], set (x−2)(x+3)=0 giving excluded values x=2 and x=−3. Excluded values must be stated even if the factor cancels during simplification.

Rational Expressions Calculator — Simplify, Add, Subtract, Multiply & Divide

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Q: How do you simplify rational expressions?

To simplify rational expressions: (1) Factor the numerator completely. (2) Factor the denominator completely. (3) Cancel any common factors. (4) State the excluded values — even factors that cancel still create excluded values. Example: (x²−4)/(x−2) = (x+2)(x−2)/(x−2) = x+2, where x ≠ 2.

Q: How do you add rational expressions with different denominators?

To add rational expressions with different denominators: (1) Factor all denominators. (2) Find the LCD — product of highest powers of all distinct factors. (3) Rewrite each fraction with the LCD as denominator. (4) Add the numerators. (5) Simplify the result. Example: 1/x + 1/(x+1) = (x+1)/[x(x+1)] + x/[x(x+1)] = (2x+1)/[x(x+1)].

Q: What is the LCD of rational expressions?

The LCD (Least Common Denominator) of rational expressions is found by: (1) factoring each denominator completely, (2) taking the highest power of each distinct factor that appears in any denominator, (3) multiplying these together. Example: LCD of x and x² is x². LCD of (x+1) and (x²−1)=(x+1)(x−1) is (x+1)(x−1).

Q: How do you divide rational expressions?

To divide rational expressions, use Keep-Change-Flip (KCF): Keep the first fraction unchanged, Change the division sign to multiplication, Flip (take the reciprocal of) the second fraction. Then multiply. Example: (x²−4)/(x+3) ÷ (x−2)/(x²−9) = (x²−4)/(x+3) × (x²−9)/(x−2). Common mistake: flipping the FIRST fraction instead of the second.

Q: Why can you only cancel factors, not terms?

Cancellation in fractions only works for factors (things connected by multiplication), NOT for terms (things connected by addition or subtraction). Example: (x²+4)/4 ≠ x² because x²+4 is a sum, not a product — you cannot cancel the 4. But (4x²)/(4) = x² because 4 is a factor of 4x². Always factor completely before cancelling.

Simplify, add, subtract, multiply, and divide rational expressions with complete step-by-step working — including LCD calculation, factoring, cross-cancellation, and excluded values for every result.

Rational Expressions Calculator

Enter the rational expression to simplify:

Fraction

Numerator

Denominator

Quick examples:

(x²−4)/(x−2)

(x²+5x+6)/(x²+3x+2)

(x³−8)/(x²−4)

(2x²+6x)/(x²+3x)

(x²−9)/(x+3)

Factoring and simplifying…

Operation:

First Fraction

Numerator

Denominator

Second Fraction

Numerator

Denominator

3/(x+2) + 2x/(x²−4)

1/x + 1/(x+1)

(x+1)/(x−3) − (x−1)/(x+3)

2/x − 3/x²

Finding LCD and combining…

First Fraction

Numerator

Denominator

Second Fraction

Numerator

Denominator

(x²−9)/(x+2) × (x²+5x+6)/(x²−x−6)

(x²−4)/(x+3) × (x+3)/(x−2)

(4x²)/(x²−4) × (x+2)/(2x)

Factoring and cross-cancelling…

First Fraction (Keep)

Numerator

Denominator

Second Fraction (Flip)

Numerator

Denominator

(x²−4)/(x+3) ÷ (x−2)/(x²−9)

(6x²)/(x+1) ÷ (3x)/(x²−1)

(x²−9)/(x+2) ÷ (x−3)/(x²+5x+6)

Applying Keep-Change-Flip…

Enter the denominator (or full expression) to find excluded values:

(x²−4)(x+3)

x²−x−6

(x²−1)(x²+x−6)

x(x+2)(x−3)

x³−4x

Solving denominator = 0…

What Is a Rational Expression? — Definition and Excluded Values

A rational expression is a fraction of the form P(x)/Q(x) where both P and Q are polynomials and Q(x) ≠ 0. This rational expressions calculator handles all five operations on rational algebraic expressions — simplifying rational expressions, adding rational expressions, subtracting, multiplying, and dividing — with full step-by-step working.

Just as rational numbers (fractions like 3/4) are formed by dividing one integer by another, rational expressions are formed by dividing one polynomial by another. Every rule that applies to arithmetic fractions also applies to rational algebraic expressions — finding a common denominator, cross-cancelling, and flipping when dividing.

Rational Expression: P(x)/Q(x) where P, Q are polynomials and Q(x) ≠ 0 Examples: (x+1)/(x−2) · (x²−4)/(x+3) · (2x²+6x)/(x²+3x)

Excluded Values (Domain Restrictions)

The excluded values of a rational expression are all values of x that make the denominator equal to zero. They are found by solving Q(x)=0. The domain is all real numbers except these excluded values.

(x+1)/(x−2) is undefined at x=2 — excluded value: x≠2
(x²+3)/x is undefined at x=0 — excluded value: x≠0
(x+1)/[(x+2)(x−3)] has excluded values: x≠−2 and x≠3

How to Simplify Rational Expressions — Step-by-Step

To simplify rational expressions, use this three-step method. The goal of simplifying rational expressions is to reduce P(x)/Q(x) to its lowest terms by cancelling all common factors.

Factor the numerator completely — use difference of squares, trinomial factoring, GCF, sum/difference of cubes as appropriate.
Factor the denominator completely — same process.
Cancel common factors — any factor that appears in both numerator and denominator can be cancelled (divided out).
State the excluded values — even factors that cancel still create excluded values from the original denominator.

Critical rule: Cancelling (x−2) from numerator and denominator does NOT remove the excluded value x=2. The original expression was undefined at x=2. Cancellation changes the algebraic form but NOT the original domain restriction. Always state x≠2 even after cancellation.

Example 1: Simplify (x²−4)/(x−2)

Factor numerator: x²−4 = (x+2)(x−2) [difference of squares]
Factor denominator: x−2 = (x−2) [already linear]
Cancel: (x+2)~~(x−2)~~/~~(x−2)~~ = x+2
State excluded value: x≠2 (denominator was zero at x=2)
Result: x+2, x≠2

Example 2: Simplify (x²+5x+6)/(x²+3x+2)

Factor numerator: x²+5x+6 = (x+2)(x+3)
Factor denominator: x²+3x+2 = (x+1)(x+2)
Cancel (x+2): result = (x+3)/(x+1)
Result: (x+3)/(x+1), x≠−1, x≠−2

Example 3: Simplify (2x²+6x)/(x²+3x)

Factor numerator: 2x²+6x = 2x(x+3)
Factor denominator: x²+3x = x(x+3)
Cancel x(x+3): result = 2
Result: 2 (constant!), x≠0, x≠−3

Adding and Subtracting Rational Expressions — The LCD Method

Adding rational expressions and subtracting rational expressions both require a common denominator. This adding rational expressions calculator uses the LCD method — the most efficient approach. Adding and subtracting rational expressions with different polynomial denominators is the most technically demanding operation.

The five-step method for adding rational expressions:

Factor all denominators completely
Find the LCD (Least Common Denominator) — product of highest powers of all distinct factors
Rewrite each fraction with the LCD as denominator (multiply numerator and denominator by the missing factors)
Add or subtract the numerators over the common LCD
Simplify the result and state excluded values

Adding Rational Expressions: 3/(x+2) + 2x/(x²−4)

Factor: x+2 = (x+2), x²−4 = (x+2)(x−2)
LCD = (x+2)(x−2)
Fraction 1: multiply by (x−2)/(x−2) → 3(x−2)/[(x+2)(x−2)]
Fraction 2: already has LCD → 2x/[(x+2)(x−2)]
Add numerators: [3(x−2)+2x]/[(x+2)(x−2)] = (3x−6+2x)/[(x+2)(x−2)] = (5x−6)/[(x+2)(x−2)]
Result: (5x−6)/[(x+2)(x−2)], x≠±2

Subtracting Rational Expressions: (x+1)/(x−3) − (x−1)/(x+3)

LCD = (x−3)(x+3)
(x+1)(x+3)/[(x−3)(x+3)] − (x−1)(x−3)/[(x−3)(x+3)]
Numerator: (x²+4x+3)−(x²−4x+3) = 8x
Result: 8x/[(x−3)(x+3)], x≠3, x≠−3

Subtraction warning: When subtracting rational expressions, the minus sign applies to the ENTIRE numerator of the second fraction, not just its first term. (x+1)/(x−3) − (x−1)/(x+3) requires expanding −(x−1)(x−3) = −(x²−4x+3), NOT just −x²+first term.

Multiplying Rational Expressions — Cross-Cancellation

The key to multiplying rational expressions is to factor everything FIRST, then cross-cancel, then multiply. This is far easier than multiplying first and then trying to simplify a large polynomial.

Cross-cancellation means cancelling factors that appear in a numerator of one fraction against factors in the denominator of EITHER fraction — not just the same fraction.

(P/Q) × (R/S) = (P·R)/(Q·S) — but FACTOR FIRST, cross-cancel, then multiply Never multiply out before factoring — you'll lose all cross-cancellation opportunities

Multiply: (x²−9)/(x+2) × (x²+5x+6)/(x²−x−6)

Factor: (x+3)(x−3)/(x+2) × (x+2)(x+3)/[(x−3)(x+2)]
Cross-cancel: (x−3) cancels with (x−3); (x+2) cancels with (x+2)
Remaining: (x+3)(x+3)/(x+2) = (x+3)²/(x+2)
Result: (x+3)²/(x+2), x≠−2, x≠3

Dividing Rational Expressions — Keep, Change, Flip

To divide rational expressions, use the Keep-Change-Flip (KCF) method. Dividing by a fraction is the same as multiplying by its reciprocal.

(P/Q) ÷ (R/S) = (P/Q) × (S/R) = (PS)/(QR) Keep first · Change ÷ to × · Flip the SECOND fraction

Common mistake: Students often flip the FIRST fraction instead of the second. Always keep the first fraction unchanged and flip only the second (divisor) fraction.

Divide: (x²−4)/(x+3) ÷ (x−2)/(x²−9)

Keep: (x²−4)/(x+3)
Change ÷ to ×
Flip second: (x²−9)/(x−2)
Multiply: (x²−4)/(x+3) × (x²−9)/(x−2)
Factor: (x+2)(x−2)/(x+3) × (x+3)(x−3)/(x−2)
Cross-cancel: (x−2) and (x+3) both cancel
Result: (x+2)(x−3) = x²−x−6, x≠±3, x≠2

Finding the LCD of Rational Expressions

The LCD (Least Common Denominator) is the most technically demanding step in adding and subtracting rational expressions. The LCD must be found algebraically by factoring — not by inspection as with simple numbers.

LCD method:

Factor each denominator completely
Identify every distinct factor across all denominators
For each distinct factor, take the highest power that appears in any single denominator
LCD = product of these highest-power factors

Denominator 1	Denominator 2	LCD
x	x²	x² (highest power of x)
(x+1)	(x+2)	(x+1)(x+2)
(x−1)²	x²−1=(x+1)(x−1)	(x+1)(x−1)²
x(x+2)	(x+2)(x−3)	x(x+2)(x−3)

Common Mistakes With Rational Expressions

Mistake 1 — Cancelling Terms Instead of Factors

❌ Wrong: (x²+4)/4 = x² (cannot cancel the 4 — x²+4 is a SUM, not a product)

✅ Correct: Only cancel factors (things connected by multiplication). (4x²)/4 = x² ✓ because 4 IS a factor of 4x².

The rule: cancel factors, not terms. You can only cancel when the common expression is multiplied throughout, not added.

Mistake 2 — Forgetting Excluded Values After Cancellation

❌ Wrong: (x²−4)/(x−2) = x+2 with no restrictions

✅ Correct: (x²−4)/(x−2) = x+2, x≠2. The excluded value x=2 must always be stated even after the factor (x−2) cancels.

Mistake 3 — Flipping the Wrong Fraction When Dividing

❌ Wrong: (A/B) ÷ (C/D) = (B/A) × (C/D)

✅ Correct: Keep the first, flip the SECOND: (A/B) ÷ (C/D) = (A/B) × (D/C)

Mistake 4 — Using Product of Denominators as LCD

Using Q·S as the denominator instead of the LCD is always valid but gives a larger expression requiring extra simplification at the end. The LCD gives the most efficient path.

Mistake 5 — Not Distributing the Negative in Subtraction

❌ Wrong: A/Q − (B+C)/S = (AS − B+C)/QS (missing the minus before C)

✅ Correct: A/Q − (B+C)/S = (AS − (B+C))/QS = (AS−B−C)/QS. The minus sign applies to the ENTIRE numerator of the second fraction.

Worked Examples — 10 Complete Problems

1. Simplify (x²−9)/(x+3)

Factor: (x+3)(x−3)/(x+3)
Cancel (x+3): x−3
Result: x−3, x≠−3

2. Simplify (2x²+x−6)/(x²+x−2)

Factor num: (2x−3)(x+2). Factor den: (x+2)(x−1)
Cancel (x+2)
Result: (2x−3)/(x−1), x≠1, x≠−2

3. Add: 3/(x+2) + 2x/(x²−4)

LCD = (x+2)(x−2)
3(x−2)/LCD + 2x/LCD = (5x−6)/[(x+2)(x−2)]
Result: (5x−6)/[(x+2)(x−2)], x≠±2

4. Subtract: (x+1)/(x−3) − (x−1)/(x+3)

LCD = (x−3)(x+3)
[(x+1)(x+3)−(x−1)(x−3)]/LCD = 8x/[(x−3)(x+3)]
Result: 8x/[(x−3)(x+3)], x≠±3

5. Add: 1/x + 1/(x+1) + 1/(x−1)

LCD = x(x+1)(x−1) = x(x²−1)
[(x+1)(x−1)+x(x−1)+x(x+1)] / [x(x+1)(x−1)]
Numerator: (x²−1)+(x²−x)+(x²+x) = 3x²−1
Result: (3x²−1)/[x(x²−1)], x≠0,±1

6. Multiply: (x²−9)/(x+2) × (x+2)²/(x−3)

Factor: (x+3)(x−3)/(x+2) × (x+2)²/(x−3)
Cross-cancel: (x−3) cancels; one (x+2) cancels
Result: (x+3)(x+2), x≠−2, x≠3

7. Multiply: (4x²)/(x²−4) × (x+2)/(2x)

Factor: 4x²/[(x+2)(x−2)] × (x+2)/(2x)
Cross-cancel: (x+2) cancels; 4x²/(2x) = 2x
Result: 2x/(x−2), x≠±2, x≠0

8. Divide: (x²−4)/(x+3) ÷ (x−2)/(x²−9)

KCF: (x²−4)/(x+3) × (x²−9)/(x−2)
Factor and cross-cancel: (x+2)(x−2)/(x+3) × (x+3)(x−3)/(x−2)
Result: (x+2)(x−3) = x²−x−6, x≠±3, x≠2

9. Divide: (6x²)/(x+1) ÷ (3x)/(x²−1)

KCF: (6x²)/(x+1) × (x²−1)/(3x)
Factor: 6x²/(x+1) × (x+1)(x−1)/(3x)
Cancel: 6x²·(x−1)/(3x) = 2x(x−1)
Result: 2x(x−1) = 2x²−2x, x≠0, x≠±1

10. Excluded values of (x+1)/[(x²−1)(x²+x−6)]

Factor: (x²−1)=(x+1)(x−1); (x²+x−6)=(x+3)(x−2)
Set each factor =0: x=−1, x=1, x=−3, x=2
Excluded values: x≠−3, −1, 1, 2

Frequently Asked Questions

What is a rational expression?

A rational expression is a fraction P(x)/Q(x) where P and Q are polynomials and Q(x) ≠ 0. Examples: (x+1)/(x−2), (x²−4)/(x+3). The domain excludes all x where Q(x)=0 — these are the excluded values.

How do you simplify rational expressions?

To simplify rational expressions: (1) Factor numerator and denominator completely. (2) Cancel common factors. (3) State excluded values — even cancelled factors create excluded values. Example: (x²−4)/(x−2) = (x+2)(x−2)/(x−2) = x+2, x≠2.

How do you add rational expressions with different denominators?

To add rational expressions: factor all denominators, find the LCD, rewrite each fraction with the LCD as denominator, add numerators, then simplify. Adding rational expressions with different polynomial denominators always requires finding the LCD first.

What is the LCD of rational expressions?

The LCD (Least Common Denominator) is found by factoring each denominator, then taking the highest power of each distinct factor across all denominators. Example: LCD of x and x² is x². LCD of (x+1) and (x²−1)=(x+1)(x−1) is (x+1)(x−1).

What are excluded values?

Excluded values are values of x that make the denominator equal zero. Set Q(x)=0 and solve. For (x+1)/[(x−2)(x+3)], excluded values are x=2 and x=−3. Always state excluded values even if the factor cancels during simplification of rational expressions.

How do you divide rational expressions?

Use Keep-Change-Flip (KCF): keep the first fraction, change ÷ to ×, flip (reciprocal of) the second fraction, then multiply. Common mistake: flipping the FIRST instead of the SECOND fraction. (A/B)÷(C/D) = (A/B)×(D/C) = AD/BC.

Why can you only cancel factors, not terms?

Cancellation works only for factors (connected by multiplication), not terms (connected by addition/subtraction). (x²+4)/4 ≠ x² because x²+4 is a sum. But 4x²/4 = x² because 4 is a factor. Always factor completely first before cancelling in rational expressions.

What is the difference between rational expressions and rational equations?

A rational expression is a fraction P(x)/Q(x) that you simplify or operate on. A rational equation sets a rational expression equal to something (e.g., (x+1)/(x−2)=3) and you solve for x by multiplying both sides by the LCD to clear fractions.

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Quick Rules

Simplify: factor → cancel → state excludedCancel factors, NOT terms

Add/Sub: find LCD → rewrite → combineFactor denominators first

Multiply: factor all → cross-cancel → multFactor BEFORE multiplying

Divide: Keep · Change · FlipFlip the SECOND fraction

LCD = highest power of each factorLeast Common Denominator

Excluded: set Q(x)=0, solve for xState even after cancellation

Try These

Simplify (x²−4)/(x−2)

Add 3/(x+2)+2x/(x²−4)

Sub (x+1)/(x−3)−(x−1)/(x+3)

Mult (x²−9)/(x+2)×...

Div (x²−4)/(x+3)÷...

Excluded (x²−4)(x+3)

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What Is a Rational Expression? — Definition and Excluded Values

Excluded Values (Domain Restrictions)

How to Simplify Rational Expressions — Step-by-Step

Example 1: Simplify (x²−4)/(x−2)

Example 2: Simplify (x²+5x+6)/(x²+3x+2)

Example 3: Simplify (2x²+6x)/(x²+3x)

Adding and Subtracting Rational Expressions — The LCD Method

Adding Rational Expressions: 3/(x+2) + 2x/(x²−4)

Subtracting Rational Expressions: (x+1)/(x−3) − (x−1)/(x+3)

Multiplying Rational Expressions — Cross-Cancellation

Multiply: (x²−9)/(x+2) × (x²+5x+6)/(x²−x−6)

Dividing Rational Expressions — Keep, Change, Flip

Divide: (x²−4)/(x+3) ÷ (x−2)/(x²−9)

Finding the LCD of Rational Expressions

Common Mistakes With Rational Expressions

Mistake 1 — Cancelling Terms Instead of Factors

Mistake 2 — Forgetting Excluded Values After Cancellation

Mistake 3 — Flipping the Wrong Fraction When Dividing

Mistake 4 — Using Product of Denominators as LCD

Mistake 5 — Not Distributing the Negative in Subtraction

Worked Examples — 10 Complete Problems

1. Simplify (x²−9)/(x+3)

2. Simplify (2x²+x−6)/(x²+x−2)

3. Add: 3/(x+2) + 2x/(x²−4)

4. Subtract: (x+1)/(x−3) − (x−1)/(x+3)

5. Add: 1/x + 1/(x+1) + 1/(x−1)

6. Multiply: (x²−9)/(x+2) × (x+2)²/(x−3)

7. Multiply: (4x²)/(x²−4) × (x+2)/(2x)

8. Divide: (x²−4)/(x+3) ÷ (x−2)/(x²−9)

9. Divide: (6x²)/(x+1) ÷ (3x)/(x²−1)

10. Excluded values of (x+1)/[(x²−1)(x²+x−6)]

Frequently Asked Questions

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Rational Expressions Calculator — Simplify, Add, Subtract, Multiply & Divide

Rational Expressions Calculator

What Is a Rational Expression? — Definition and Excluded Values

Excluded Values (Domain Restrictions)

How to Simplify Rational Expressions — Step-by-Step

Example 1: Simplify (x²−4)/(x−2)

Example 2: Simplify (x²+5x+6)/(x²+3x+2)

Example 3: Simplify (2x²+6x)/(x²+3x)

Adding and Subtracting Rational Expressions — The LCD Method

Adding Rational Expressions: 3/(x+2) + 2x/(x²−4)

Subtracting Rational Expressions: (x+1)/(x−3) − (x−1)/(x+3)

Multiplying Rational Expressions — Cross-Cancellation

Multiply: (x²−9)/(x+2) × (x²+5x+6)/(x²−x−6)

Dividing Rational Expressions — Keep, Change, Flip

Divide: (x²−4)/(x+3) ÷ (x−2)/(x²−9)

Finding the LCD of Rational Expressions

Common Mistakes With Rational Expressions

Mistake 1 — Cancelling Terms Instead of Factors

Mistake 2 — Forgetting Excluded Values After Cancellation

Mistake 3 — Flipping the Wrong Fraction When Dividing

Mistake 4 — Using Product of Denominators as LCD

Mistake 5 — Not Distributing the Negative in Subtraction

Worked Examples — 10 Complete Problems

1. Simplify (x²−9)/(x+3)

2. Simplify (2x²+x−6)/(x²+x−2)

3. Add: 3/(x+2) + 2x/(x²−4)

4. Subtract: (x+1)/(x−3) − (x−1)/(x+3)

5. Add: 1/x + 1/(x+1) + 1/(x−1)

6. Multiply: (x²−9)/(x+2) × (x+2)²/(x−3)

7. Multiply: (4x²)/(x²−4) × (x+2)/(2x)

8. Divide: (x²−4)/(x+3) ÷ (x−2)/(x²−9)

9. Divide: (6x²)/(x+1) ÷ (3x)/(x²−1)

10. Excluded values of (x+1)/[(x²−1)(x²+x−6)]

Frequently Asked Questions

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