Determine equivalent rational expressions and non-permissible values

Alberta Grade 12 Mathematics

Relations and Functions

How factoring reveals when two rational expressions have the same value

A rational expression is a fraction whose numerator and denominator are polynomials. Rational expressions can look different but have the same value for all inputs they both allow. The denominator matters: division by zero is undefined. In this lesson, you will practise factoring, comparing expressions, and keeping track of values that the original expression does not permit.

What you will learn

Identify values that make a rational expression undefined.
Factor polynomials to compare rational expressions.
Determine whether two rational expressions are equivalent.
State non-permissible values from the original denominators, including when a factor cancels.

1. Review: fractions, factors, and zero denominators

A fraction is undefined when its denominator is zero. For example, a fraction with denominator

x-4

cannot use

x=4

, because that makes the denominator zero.

A factor is an expression multiplied by another expression. For instance,

x(x+3)

has factors

x

and

x+3

. Factoring rewrites a polynomial as a product. This is useful because a product is zero when at least one factor is zero.

To find the values that make a denominator zero, set each denominator factor equal to zero and solve. These excluded inputs are called non-permissible values. They are excluded from the original rational expression’s domain, meaning the set of inputs for which the expression is defined.

\frac{a}{0}

is undefined

Never allow a denominator to equal zero.
Factor a denominator before finding its non-permissible values.

2. Equivalent expressions and restrictions

Equivalent rational expressions have the same value for every input that is allowed in both expressions. Factoring can show that two forms are equivalent. For example, multiplying or dividing the numerator and denominator by the same nonzero factor does not change the fraction’s value.

A factor may cancel from the numerator and denominator. Cancellation does not make its zero value permissible. The original denominator still determines which inputs were excluded.

For example, the expression with numerator

(x-2)(x+5)

and denominator

x-2

simplifies to

x+5

, but the original expression still excludes

x=2

. The simplified expression is defined there; the original one is not. Therefore, when stating an equivalent form, include the restrictions from the original expression.

\frac{A\cdot C}{B\cdot C}=\frac{A}{B},\quad B\ne 0,\ C\ne 0

Factor first, then identify denominator restrictions.
Cancel common factors only after factoring.
Keep restrictions from the original denominator, even after cancellation.

3. A reliable method for comparing rational expressions

Start by finding the non-permissible values of each expression from its original denominator. Then factor each numerator and denominator completely at the level needed to reveal common factors.

After factoring, cancel only factors that appear in both the numerator and denominator of the same fraction. To decide if two expressions are equivalent, compare their factored forms after cancellation, while retaining the restrictions from both original expressions.

You can also compare two fractions by cross-multiplying their numerators and denominators, provided you retain the original restrictions. Factoring is often clearer because it displays common factors and excluded values directly. No method makes a zero denominator permissible.

\frac{P}{Q}=\frac{R}{S}\quad\Longleftrightarrow\quad PS=RQ,\quad Q\ne0,\ S\ne0

Record restrictions before simplifying.
A cancelled factor still gives a non-permissible value.
Two expressions can be equivalent on their shared domain even if their displayed forms differ.

4. Use the restrictions as part of your answer

A complete answer names both the simplified or equivalent expression and the non-permissible values. This matters because the algebraic form alone may hide an excluded input.

When a denominator is already factored, its restrictions are easy to see. When it is not, factor it first. For a quadratic denominator, find factors whose product gives the constant term and whose sum gives the coefficient of the middle term, or use another familiar factoring method.

Before finishing, check that every stated restriction makes an original denominator zero and that no zero of an original denominator has been missed.

Restrictions belong to the original expression, not only the final simplified form.
Substitute each excluded value into the original denominator as a check.

Worked example

Cancel a common factor and retain its restriction

Simplify

\frac{x^2-9}{x-3}

and state the non-permissible value.

Find the restriction
The original denominator is $x-3$ . It is zero when $x=3$ , so that input is not allowed.
$x-3=0\Rightarrow x=3$
Factor the numerator
The numerator is a difference of squares: the difference between two squares factors as the difference of their square roots times their sum.
$x^2-9=(x-3)(x+3)$
Cancel and state the result
The common factor cancels for inputs where it is nonzero. Keep the original restriction, because the starting expression was undefined at $x=3$ .
$\frac{(x-3)(x+3)}{x-3}=x+3,\quad x\ne3$

Answer: The equivalent simplified expression is

x+3

, with the restriction

x\ne3

.

Check: At

x=3

, the original denominator is zero. The simplified expression gives a value there, but that does not change the original restriction.

Worked example

Determine whether two forms are equivalent

Are

\frac{x^2+5x+6}{x+2}

and

x+3

equivalent? State all restrictions from the displayed expressions.

Find restrictions
The fraction’s denominator is zero when $x=-2$ . The expression $x+3$ has no denominator, so it has no restriction of this type.
$x+2=0\Rightarrow x=-2$
Factor the numerator
The two numbers with product $6$ and sum $5$ are $2$ and $3$ , so the quadratic factors as shown.
$x^2+5x+6=(x+2)(x+3)$
Compare the expressions
After cancelling the common factor, the fraction has the form $x+3$ whenever $x\ne-2$ . Thus the expressions agree on the fraction’s allowed inputs. The fraction still excludes $-2$ .
$\frac{(x+2)(x+3)}{x+2}=x+3,\quad x\ne-2$

Answer: They are equivalent for

x\ne-2

. The fraction’s non-permissible value is

x=-2

; the polynomial expression has no denominator restriction.

Check: At

x=0

, both expressions equal

3

. At

x=-2

, the fraction is undefined, so equivalence does not mean that both expressions are defined for every real number.

Worked example

Compare expressions with different denominators

Determine whether

\frac{x^2-4}{x^2+x-6}

and

\frac{x+2}{x+3}

are equivalent, and state the non-permissible values.

Find the restrictions from both denominators
Factor the first denominator and set each factor to zero. The second denominator is zero at $x=-3$ , which is already among the exclusions.
$x^2+x-6=(x+3)(x-2),\quad x=-3,\ 2$
Factor the first numerator
The numerator is a difference of squares. Rewrite the first fraction using its factors.
$x^2-4=(x-2)(x+2)$
Compare the simplified forms
The common factor $x-2$ cancels, leaving the same form as the second expression. Both must still exclude every value that made either original denominator zero.
$\frac{(x-2)(x+2)}{(x+3)(x-2)}=\frac{x+2}{x+3},\quad x\ne2,-3$

Answer: The expressions are equivalent for

x\ne2,-3

. The non-permissible values are

x=2

and

x=-3

.

Check: The first denominator is zero at both excluded values. The second denominator is zero at

x=-3

. Keeping both exclusions accounts for the original expressions.

Common mistakes and how to avoid them

Cancelling a factor and then allowing the value that makes it zero.

Correction: Find restrictions from the original denominator first. Keep them after simplifying.

Finding non-permissible values from the numerator.

Correction: Only denominator values can make a rational expression undefined through division by zero.

Saying two expressions are equivalent without noting where they are defined.

Correction: State the values excluded by the original denominators, and say the expressions agree on their shared allowed inputs.

Lesson summary

A rational expression is undefined when its denominator is zero.
Factor denominators to find non-permissible values.
Factor before cancelling common factors.
Cancellation does not restore an input excluded by an original denominator.
To compare expressions, simplify carefully and retain restrictions from both original expressions.

Check your understanding

Question 1

Simplify

\frac{x^2-16}{x-4}

and state its restriction.

$x+4$ , with $x\ne4$
$x+4$ , with no restriction
$x-4$ , with $x\ne4$
$x+4$ , with $x\ne-4$

Show answer and explanation

x+4

, with

x\ne4

Factor the numerator as

(x-4)(x+4)

. The factor cancels, but the original denominator still excludes

x=4

.

Question 2

Which values are non-permissible for

\frac{x+1}{(x-5)(x+2)}

?

$x=5$ and $x=-2$
$x=-1$ only
$x=5$ only
$x=-2$ and $x=-1$

Show answer and explanation

x=5

and

x=-2

Set each denominator factor equal to zero. The numerator does not determine non-permissible values.

Question 3

Are

\frac{x^2-1}{x-1}

and

x+1

equivalent?

Yes, for $x\ne1$
Yes, for all real values of $x$
No, because the expressions look different
No, because $x=-1$ is excluded

Show answer and explanation

Yes, for

x\ne1

The numerator factors as

(x-1)(x+1)

. The common factor cancels, but the original fraction excludes

x=1

.

Key terms

Rational expression: A fraction whose numerator and denominator are polynomials.
Factor: One of the expressions multiplied together to form a product.
Equivalent rational expressions: Rational expressions that have the same value for every input allowed in both.
Non-permissible value: An input that makes the original denominator zero, so the expression is undefined.
Domain: The set of inputs for which an expression is defined.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation RF1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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RF1 · Determine equivalent rational expressions and non-permissible values

Alberta Grade 12 Mathematics

How factoring reveals when two rational expressions have the same value

What you will learn

1. Review: fractions, factors, and zero denominators

2. Equivalent expressions and restrictions

3. A reliable method for comparing rational expressions

4. Use the restrictions as part of your answer

Cancel a common factor and retain its restriction

Determine whether two forms are equivalent

Compare expressions with different denominators

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-2

About this lesson and its review