Simplifying Algebraic Fractions

Simplifying an algebraic fraction means rewriting it in an equivalent form with no common factors remaining in the numerator and denominator.

For example:

6x/9x

can be simplified because the numerator and denominator share common factors.

Factor:

6x/9x = (3x × 2)/(3x × 3)

Cancel the common factor 3x:

2/3

Therefore:

6x/9x = 2/3, where x ≠ 0

The most important rule is:

Cancel factors, not terms.

This single idea prevents many of the most common mistakes with algebraic fractions.

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Start by Finding Restrictions

Before simplifying, identify values that make the original denominator equal to zero.

Consider:

(x² + 3x)/x

The denominator is x.

Therefore:

x ≠ 0

Now simplify.

Factor the numerator:

x(x + 3)/x

Cancel x:

x + 3

Final answer:

x + 3, where x ≠ 0

The restriction remains even though x disappeared from the denominator after simplification.

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A Reliable Method

Use:

Restrict → Factor → Cancel → Simplify → Verify

Restrict

Find values that make the original denominator zero.

Factor

Factor the numerator and denominator completely.

Cancel

Cancel factors that appear in both numerator and denominator.

Simplify

Write the remaining expression clearly.

Verify

Check that the simplified expression gives the same result as the original for an allowed value.

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Factors and Terms

Understanding the difference between factors and terms is essential.

Consider:

3x(x + 4)

The factors are:

• 3
• x
• x + 4

Now consider:

3x + 12

The terms are:

• 3x
• 12

Terms are separated by addition or subtraction.

Factors are multiplied together.

The cancellation rule

You may cancel common factors.

You may not simply cancel terms.

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Numerical Fractions First

The same principle works with ordinary fractions.

Consider:

18/24

Factor:

(6 × 3)/(6 × 4)

Cancel 6:

3/4

Algebraic fractions use exactly the same idea.

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Simplifying Numerical Coefficients

Simplify:

12x/18

The greatest common factor of 12 and 18 is 6.

Factor:

(6 × 2x)/(6 × 3)

Cancel:

2x/3

Therefore:

12x/18 = 2x/3

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Cancelling Variable Factors

Simplify:

8x/4x

Restriction:

x ≠ 0

Factor:

(4x × 2)/(4x)

Cancel 4x:

2

Therefore:

8x/4x = 2, where x ≠ 0

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Powers of Variables

Simplify:

x⁴/x²

Restriction:

x ≠ 0

Expand the factors:

(x × x × x × x)/(x × x)

Cancel two x factors:

x²

Therefore:

x⁴/x² = x², where x ≠ 0

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Using Exponent Rules

When the same base appears in the numerator and denominator:

xᵐ/xⁿ = xᵐ⁻ⁿ

when appropriate and x ≠ 0.

For example:

x⁷/x³ = x⁴

because:

7 − 3 = 4

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Coefficients and Powers Together

Simplify:

18x⁵/12x²

First simplify:

18/12 = 3/2

Then:

x⁵/x² = x³

Therefore:

18x⁵/12x² = 3x³/2

Restriction:

x ≠ 0

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Multiple Variables

Simplify:

12x²y/18xy²

Restrictions:

x ≠ 0

y ≠ 0

Simplify the numbers:

12/18 = 2/3

Simplify x:

x²/x = x

Simplify y:

y/y² = 1/y

Therefore:

12x²y/18xy² = 2x/(3y)

with:

x ≠ 0, y ≠ 0

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Factor Before Cancelling

Many algebraic fractions cannot be simplified until the numerator or denominator has been factored.

Consider:

(3x + 12)/3

At first, the numerator contains two terms.

Factor out 3:

3(x + 4)/3

Now cancel:

x + 4

Therefore:

(3x + 12)/3 = x + 4

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Factoring Out a Variable

Simplify:

(x² + 6x)/x

Restriction:

x ≠ 0

Factor x from the numerator:

x(x + 6)/x

Cancel:

x + 6

Therefore:

x + 6, where x ≠ 0

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Factoring Numbers and Variables

Simplify:

(6x² + 12x)/(6x)

Restriction:

x ≠ 0

Factor the numerator:

6x(x + 2)/(6x)

Cancel 6x:

x + 2

Therefore:

x + 2, where x ≠ 0

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Factoring the Denominator

Sometimes the denominator needs to be factored.

Simplify:

3x/(3x + 6)

Factor the denominator:

3x/[3(x + 2)]

The numerator can be written:

3x

Cancel 3:

x/(x + 2)

Restriction from the original denominator:

3x + 6 ≠ 0

Therefore:

x ≠ −2

Final answer:

x/(x + 2), where x ≠ −2

Notice that x itself does not cancel because x + 2 is a factor, not x.

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Factoring Both Numerator and Denominator

Simplify:

(2x + 6)/(4x + 12)

Factor both:

2(x + 3)/[4(x + 3)]

Restriction:

x ≠ −3

Cancel x + 3:

2/4

Simplify:

1/2

Therefore:

(2x + 6)/(4x + 12) = 1/2, where x ≠ −3

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Common Binomial Factors

A complete expression such as:

x + 5

can be a factor.

Consider:

3(x + 5)/[7(x + 5)]

Restriction:

x ≠ −5

Cancel the common factor x + 5:

3/7

Therefore:

3/7, where x ≠ −5

The entire binomial must match.

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Difference of Squares

An important factoring pattern is:

a² − b² = (a − b)(a + b)

For example:

x² − 9

factors as:

(x − 3)(x + 3)

This pattern is very useful when simplifying algebraic fractions.

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Example Using Difference of Squares

Simplify:

(x² − 9)/(x − 3)

Restriction:

x ≠ 3

Factor the numerator:

(x − 3)(x + 3)/(x − 3)

Cancel:

x + 3

Therefore:

x + 3, where x ≠ 3

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Difference of Squares in the Denominator

Simplify:

(x − 4)/(x² − 16)

Factor the denominator:

x² − 16 = (x − 4)(x + 4)

So:

(x − 4)/[(x − 4)(x + 4)]

Cancel x − 4:

1/(x + 4)

But restrictions come from the original denominator.

Original denominator:

(x − 4)(x + 4)

Therefore:

x ≠ 4

and:

x ≠ −4

Final answer:

1/(x + 4), where x ≠ ±4

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Factoring Simple Trinomials

Consider:

x² + 5x + 6

We need two numbers that:

• multiply to 6
• add to 5

Those numbers are 2 and 3.

Therefore:

x² + 5x + 6 = (x + 2)(x + 3)

This allows more algebraic fractions to be simplified.

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Simplifying with a Trinomial

Simplify:

(x² + 5x + 6)/(x + 2)

Restriction:

x ≠ −2

Factor:

(x + 2)(x + 3)/(x + 2)

Cancel:

x + 3

Therefore:

x + 3, where x ≠ −2

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Factoring Numerator and Denominator Completely

Simplify:

(x² + 5x + 6)/(x² + 4x + 3)

Factor the numerator:

x² + 5x + 6 = (x + 2)(x + 3)

Factor the denominator:

x² + 4x + 3 = (x + 1)(x + 3)

Restrictions:

x ≠ −1

x ≠ −3

Now:

[(x + 2)(x + 3)]/[(x + 1)(x + 3)]

Cancel x + 3:

(x + 2)/(x + 1)

Final answer:

(x + 2)/(x + 1), where x ≠ −1, −3

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Why You Must Factor Completely

Consider:

(x² − 4)/(x² + 3x + 2)

Without factoring, no common factor is obvious.

Factor the numerator:

x² − 4 = (x − 2)(x + 2)

Factor the denominator:

x² + 3x + 2 = (x + 1)(x + 2)

Now:

[(x − 2)(x + 2)]/[(x + 1)(x + 2)]

Cancel x + 2:

(x − 2)/(x + 1)

Restrictions:

x ≠ −1, −2

Factoring reveals the common factor.

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When Nothing Cancels

Not every algebraic fraction can be simplified.

Consider:

(x + 3)/(x + 5)

The numerator is one factor:

x + 3

The denominator is another:

x + 5

They are different factors.

Therefore nothing cancels.

The fraction is already in simplest form.

Restriction:

x ≠ −5

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Another Expression That Cannot Be Simplified

Consider:

(2x + 3)/(x + 4)

Neither expression has a common factor.

Therefore:

(2x + 3)/(x + 4)

is already simplified.

Restriction:

x ≠ −4

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Do Not Force Simplification

Students sometimes assume that every algebraic fraction must become shorter.

That is not true.

For example:

(x + 7)/(x + 2)

cannot be simplified.

Recognizing that an expression is already simplified is an important algebraic skill.

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The Most Common Cancellation Error

Consider:

(x + 4)/x

Incorrect reasoning:

Cancel x.

This would be wrong because x is only part of one term in the numerator.

The numerator is:

x + 4

not:

x × 4

Therefore nothing cancels.

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Why Cancellation Across Addition Is Wrong

Suppose x = 2.

Original expression:

(2 + 4)/2 = 6/2 = 3

If we incorrectly cancelled x and claimed:

(x + 4)/x = 4

we would get:

4

But:

3 ≠ 4

Therefore the cancellation is invalid.

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Cancellation Across Subtraction Is Also Wrong

Consider:

(x − 5)/x

You cannot cancel x.

The numerator consists of two terms:

x

and:

−5

There is no common x factor multiplying the entire numerator.

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Correct Way to Look for Cancellation

Ask:

Can the entire numerator and denominator be written as products?

For:

(x² + 5x)/x

yes:

x(x + 5)/x

Now x is a common factor and can be cancelled.

For:

(x + 5)/x

the numerator cannot be factored as x times another simple polynomial expression.

So nothing cancels.

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Cancelling Numerical Parts Correctly

Consider:

(6x + 12)/(9x + 18)

It would be incorrect to cancel 6 with 9 immediately while ignoring the rest of each expression.

Instead, factor:

Numerator:

6(x + 2)

Denominator:

9(x + 2)

Then:

6(x + 2)/[9(x + 2)]

Cancel x + 2:

6/9

Simplify:

2/3

Restriction:

x ≠ −2

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Negative Common Factors

Consider:

(x − 3)/(3 − x)

Notice:

3 − x = −(x − 3)

Therefore:

(x − 3)/[−(x − 3)]

Cancel x − 3:

−1

Restriction:

x ≠ 3

Therefore:

(x − 3)/(3 − x) = −1, where x ≠ 3

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Reversed Binomial Factors

This pattern is useful:

a − b = −(b − a)

For example:

x − 5 = −(5 − x)

So expressions that look almost identical may differ by a factor of −1.

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Example with Reversed Factors

Simplify:

(x + 2)/(−x − 2)

Factor −1 from the denominator:

−x − 2 = −(x + 2)

So:

(x + 2)/[−(x + 2)]

Cancel:

−1

Restriction:

x ≠ −2

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Multiple Common Factors

Simplify:

6x(x + 4)/[9x²(x + 4)]

Restrictions:

x ≠ 0

x ≠ −4

Simplify 6/9:

2/3

Simplify:

x/x² = 1/x

Cancel x + 4.

Result:

2/(3x)

Therefore:

6x(x + 4)/[9x²(x + 4)] = 2/(3x)

where:

x ≠ 0, −4

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A More Challenging Example

Simplify:

(2x² − 8)/(x² + 3x + 2)

Factor the numerator:

2(x² − 4)

Then use difference of squares:

2(x − 2)(x + 2)

Factor the denominator:

(x + 1)(x + 2)

Restrictions:

x ≠ −1, −2

Now:

2(x − 2)(x + 2)/[(x + 1)(x + 2)]

Cancel x + 2:

2(x − 2)/(x + 1)

Final answer:

2(x − 2)/(x + 1), where x ≠ −1, −2

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Another Challenging Example

Simplify:

(x² − 6x + 9)/(x² − 9)

Factor the numerator:

x² − 6x + 9 = (x − 3)²

Factor the denominator:

x² − 9 = (x − 3)(x + 3)

Restrictions:

x ≠ 3, −3

Then:

(x − 3)(x − 3)/[(x − 3)(x + 3)]

Cancel one x − 3:

(x − 3)/(x + 3)

Final answer:

(x − 3)/(x + 3), where x ≠ ±3

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Verifying a Simplified Answer

There are several ways to check simplification.

Factor Check

Make sure the numerator and denominator have been completely factored and no common factors remain.

Reverse Check

Multiply cancelled factors back into the simplified expression.

Numerical Check

Choose an allowed value for the variable and evaluate both the original and simplified expressions.

The results should agree.

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Numerical Verification Example

Suppose:

(x² − 9)/(x − 3)

was simplified to:

x + 3

with:

x ≠ 3

Choose:

x = 5

Original:

(5² − 9)/(5 − 3)

= (25 − 9)/2

= 16/2

= 8

Simplified:

5 + 3 = 8

The values agree.

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A Check Can Reveal an Error

Suppose someone incorrectly claims:

(x + 4)/x = 4

Choose:

x = 2

Original:

(2 + 4)/2 = 3

Claimed simplified expression:

4

Since:

3 ≠ 4

the simplification is incorrect.

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Restrictions Must Also Be Verified

Suppose:

(x² − 4)/(x − 2)

simplifies to:

x + 2

A student might write only:

x + 2

But the original expression is undefined at:

x = 2

So the complete simplified form is:

x + 2, where x ≠ 2

Verification includes preserving the original restrictions.

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Simplified Expressions and Graphs

Consider:

y = (x² − 4)/(x − 2)

After factoring:

y = [(x − 2)(x + 2)]/(x − 2)

So:

y = x + 2, where x ≠ 2

The graph looks like the straight line:

y = x + 2

except that the point corresponding to x = 2 is missing.

At x = 2:

y = 4

would normally lie on the line, but it is excluded from the original expression.

This creates a hole in the graph.

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Simplifying Science Formulas

Algebraic fractions occur frequently in science.

For example:

ρ = m/V

If both mass and volume contain a common conversion factor, simplifying before calculating can make the arithmetic easier.

Other examples include:

v = d/t

p = F/A

R = V/I

P = E/t

The same mathematical rules apply: denominators cannot be zero, and common factors may be simplified where appropriate.

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Simplifying Units

The idea of cancellation also applies to units.

For example:

v = 100 m / 20 s

Simplify:

v = 5 m/s

In more advanced science calculations, units themselves can behave like algebraic factors.

For example:

(kg·m/s²) × m

becomes:

kg·m²/s²

which is equivalent to a joule.

This is one reason strong algebraic simplification skills are useful in science.

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Efficient Simplification

Before doing a lot of expansion, look for factorization.

For example:

(x + 4)(x − 2)/[(x + 4)(x + 7)]

is already factored.

Simply cancel:

x + 4

to obtain:

(x − 2)/(x + 7)

Restrictions from the original denominator:

x ≠ −4, −7

Expanding everything first would make the problem unnecessarily difficult.

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Do Not Expand When Factoring Is More Useful

Consider:

(x − 3)(x + 5)/(x − 3)

The common factor is immediately visible.

Cancel:

x + 5

with:

x ≠ 3

Expanding the numerator to:

x² + 2x − 15

would hide the common factor.

When simplifying algebraic fractions, factored form is usually more useful than expanded form.

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A Full Worked Example

Simplify:

(3x² − 12x)/(6x² − 24)

Restrict

Factor the original denominator:

6x² − 24 = 6(x² − 4)

= 6(x − 2)(x + 2)

Therefore:

x ≠ 2, −2

Factor the numerator

3x² − 12x = 3x(x − 4)

So:

3x(x − 4)/[6(x − 2)(x + 2)]

Cancel

The coefficients 3 and 6 share a factor of 3:

x(x − 4)/[2(x − 2)(x + 2)]

There are no other common factors.

Therefore:

x(x − 4)/[2(x − 2)(x + 2)]

with:

x ≠ ±2

The fact that the final expression is still a fraction does not mean we have failed to simplify it. No common factors remain.

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Recognizing the Simplest Form

An algebraic fraction is generally simplified when:

• numerator and denominator are fully factored where useful
• numerical coefficients are reduced
• no common factors remain
• unnecessary factors of 1 are removed
• restrictions are stated when relevant

The result does not need to be a single term.

For example:

(x + 2)/(x − 5)

may already be completely simplified.

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Common Mistakes

Mistake: Cancelling terms instead of factors.

Incorrect:

(x + 5)/x → 5

Correct:

Nothing cancels.

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Mistake: Cancelling across addition.

Incorrect:

(2x + 6)/2x → 3

Factor first.

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Mistake: Cancelling across subtraction.

Incorrect:

(x − 4)/x → −4

Nothing cancels.

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Mistake: Forgetting to factor.

For:

(x² + 5x)/x

factor first:

x(x + 5)/x

then simplify:

x + 5

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Mistake: Cancelling only part of a factor.

If the factor is:

x + 3

you may cancel it only with another complete factor x + 3.

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Mistake: Treating x + 3 and x − 3 as common factors.

They are different expressions and cannot cancel.

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Mistake: Treating x − 3 and 3 − x as identical.

They differ by a factor of −1.

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Mistake: Forgetting restrictions.

Restrictions come from the original denominator and remain after simplification.

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Mistake: Assuming every fraction can be simplified.

Some algebraic fractions already have no common factors.

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Mistake: Expanding an already factored expression.

Expansion can hide the factors needed for cancellation.

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Mistake: Checking a result using a restricted value.

Choose an allowed value when verifying by substitution.

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Key Terms

Algebraic fraction — fraction containing algebraic expressions in the numerator, denominator, or both.

Numerator — expression above the fraction line.

Denominator — expression below the fraction line.

Factor — quantity multiplied by another quantity.

Term — part of an expression separated by addition or subtraction.

Common factor — factor shared by the numerator and denominator.

Greatest common factor (GCF) — greatest factor shared by two or more expressions.

Factorization — rewriting an expression as a product of factors.

Simplify — rewrite an expression in an equivalent form with unnecessary common factors removed.

Cancel — divide numerator and denominator by the same nonzero factor.

Restriction — variable value excluded because it makes the original denominator zero.

Difference of squares — pattern a² − b² = (a − b)(a + b).

Trinomial — polynomial containing three terms.

Equivalent expressions — expressions with the same value throughout their shared permissible domain.

Verification — process of checking that a mathematical result is correct.

Removable discontinuity — missing point in the graph caused when a common factor cancels but the original restriction remains.

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Key Takeaways

• Simplifying algebraic fractions is based on the same principles as simplifying numerical fractions.
• Find restrictions from the original denominator before simplifying.
• Factor the numerator and denominator completely where possible.
• The most important rule is cancel factors, not terms.
• Addition and subtraction separate terms.
• Multiplication creates factors.
• Numerical common factors can be cancelled.
• Variable common factors can be cancelled.
• Entire binomial factors can be cancelled.
• Expressions such as x + 3 can cancel only with an identical factor x + 3.
• Reversed factors such as x − 3 and 3 − x differ by −1.
• Difference-of-squares factoring can reveal common factors.
• Factoring trinomials can also reveal common factors.
• Restrictions remain even if the factor causing the restriction is cancelled.
• Not every algebraic fraction can be simplified.
• A fraction is simplified when no common factors remain.
• Factored form is usually more useful than expanded form when simplifying fractions.
• Expanding too early can hide common factors.
• Numerical substitution can verify a simplified result.
• A valid verification value must not violate the original restrictions.
• Graphically, a cancelled factor may produce a hole in a rational function.
• Algebraic fraction simplification is important for later work with rational equations, functions, algebraic manipulation, and science formulas.

Check Your Understanding

1. Explain the difference between a factor and a term.
2. Simplify 12x/18.
3. Simplify 15x/5x and state the restriction.
4. Simplify 18x⁴/12x² and state the restriction.
5. Simplify 12x²y/8xy and state any restrictions.
6. Simplify (4x + 12)/4.
7. Simplify (x² + 7x)/x and state the restriction.
8. Simplify (6x² + 18x)/(6x) and state the restriction.
9. Explain why x cannot be cancelled from (x + 5)/x.
10. Determine whether (x + 4)/(x + 7) can be simplified. Explain your reasoning.
11. Simplify (x² − 16)/(x − 4) and state the restriction.
12. Simplify (x − 5)/(x² − 25) and state all restrictions.
13. Simplify (x² + 6x + 8)/(x + 2) and state the restriction.
14. Simplify (x² + 5x + 6)/(x² + 4x + 3) and state all restrictions.
15. Simplify (2x + 8)/(3x + 12) and state the restriction.
16. Simplify (x − 6)/(6 − x) and state the restriction.
17. A student claims that (x + 3)/x = 3 after cancelling x. Explain the error.
18. Explain why restrictions must be found from the original denominator rather than only from the simplified expression.
19. Describe how substitution can be used to verify that two algebraic expressions are equivalent.
20. Simplify (2x² − 18)/[(x − 3)(x + 4)], state all restrictions, and explain each step of your simplification.