Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Rational Expressions: Simplifying and Stating Restrictions

A3.3About 70 minutes
Learning goals
  • I can explain what a rational expression is.
  • I can find the restrictions on the variable in a rational expression.
  • I can simplify a rational expression by factoring and dividing out common factors.
  • I can explain why only factors, not terms, can be divided out.

Words to know

rational expression
A fraction whose numerator and denominator are polynomials, such as (x − 3)/(x + 2).
restriction
A value of the variable that is not allowed because it makes a denominator zero.
factor (noun)
Something multiplied: in 2x(x − 4), the factors are 2, x and (x − 4).
term
Something added or subtracted: in x + 4, the terms are x and 4.
opposites
Expressions such as a − b and b − a, whose quotient is −1.

Fractions with variables

You simplify 12/18 to 2/3 by dividing the top and bottom by their common factor 6. A rational expression is a fraction whose numerator and denominator are polynomials, such as

(x² − 9)/(x² + 5x + 6)

It is simplified the same way: factor, then divide out common factors. There is one new rule: because we can never divide by zero, we must state which values of the variable are not allowed. These are called restrictions.

Rational expressions show up in rates (time = distance ÷ speed), averages and, in Grade 12 (MHF4U), in rational functions.

Remember from Grade 10

In Grade 10 you factored using a common factor, trinomials x² + bx + c and ax² + bx + c, differences of squares a² − b² = (a − b)(a + b), and perfect squares. Every simplification in this lesson starts with these.

Restrictions

A restriction is any value of the variable that makes a denominator zero. To find them:

  1. Factor the denominator.
  2. Set each factor equal to 0 and solve.
  3. Write the restrictions, for example x ≠ −2, x ≠ −3.

State restrictions from the original expression, before you cancel anything. A factor that cancels still made the original denominator zero.

Simplifying

  1. Factor the numerator and denominator completely.
  2. State the restrictions (from the denominator).
  3. Divide out common factors.
  4. Write the simplified expression with its restrictions.
Worked example

Example 1: Monomials

Simplify 12x³y / (18xy²) and state the restrictions.

  1. Restrictions: the denominator 18xy² is 0 if x = 0 or y = 0. So x ≠ 0, y ≠ 0.
  2. Numbers: 12/18 = 2/3. Powers: x³/x = x² and y/y² = 1/y (subtract exponents).
  3. Result: 2x²/(3y), x ≠ 0, y ≠ 0.

2x²/(3y), x ≠ 0, y ≠ 0

Worked example

Example 2: Trinomials

Simplify (x² − 9)/(x² + 5x + 6) and state the restrictions.

  1. Factor: numerator (x − 3)(x + 3); denominator (x + 2)(x + 3).
  2. Restrictions: x + 2 = 0 or x + 3 = 0, so x ≠ −2, x ≠ −3.
  3. Divide out (x + 3): (x − 3)/(x + 2).
  4. Notice x ≠ −3 is still a restriction even though (x + 3) no longer appears.

(x − 3)/(x + 2), x ≠ −2, −3

xy−6−224−10−8−4−22(−3, −6): holey = x − 3, x ≠ −3y = (x² − 9)/(x + 3)= (x − 3)(x + 3)/(x + 3)= x − 3, x ≠ −3At x = −3 the originalgives 0/0: undefined.The graph has a hole there.
(x² − 9)/(x + 3) and x − 3 agree everywhere except at x = −3, where the original is undefined. That missing point is why restrictions are kept after simplifying.
Worked example

Example 3: Common factor and difference of squares

Simplify (2x² − 8x)/(x² − 16).

  1. Factor: 2x(x − 4) over (x − 4)(x + 4).
  2. Restrictions: x ≠ 4, x ≠ −4.
  3. Divide out (x − 4): 2x/(x + 4), x ≠ ±4.

2x/(x + 4), x ≠ ±4

Worked example

Example 4: Opposites

Simplify (5 − x)/(x² − 25).

  1. The denominator is (x − 5)(x + 5), so x ≠ ±5.
  2. 5 − x looks like x − 5 but backwards. Factor out −1: 5 − x = −(x − 5).
  3. So the expression is −(x − 5) / [(x − 5)(x + 5)] = −1/(x + 5), x ≠ ±5.
  4. Check with x = 0: original 5/(−25) = −0.2; answer −1/5 = −0.2 ✓.

−1/(x + 5), x ≠ ±5

Opposite factors

(a − b) and (b − a) are opposites: (b − a) = −(a − b). Their quotient is −1. Spot them by checking whether the terms are the same with the order switched.

Worked example

Example 5: Trinomials with a ≠ 1

Simplify (6x² − x − 2)/(4x² − 1).

  1. Factor the numerator: find two numbers with product 6 × (−2) = −12 and sum −1: −4 and 3. 6x² − 4x + 3x − 2 = 2x(3x − 2) + 1(3x − 2) = (3x − 2)(2x + 1).
  2. Factor the denominator (difference of squares): (2x − 1)(2x + 1).
  3. Restrictions: 2x − 1 = 0 gives x = 1/2; 2x + 1 = 0 gives x = −1/2. So x ≠ ±1/2.
  4. Divide out (2x + 1): (3x − 2)/(2x − 1).

(3x − 2)/(2x − 1), x ≠ ±1/2

YES: cancel a common FACTOR(x + 3)(x − 3)────────────(x + 3)(x + 1)= (x − 3)/(x + 1)NO: you cannot cancel TERMS(x + 3)/x ≠ 3test x = 2:(2 + 3)/2 = 2.5, not 3x is a term of x + 3, not a factor
Only factors (things multiplied) can be divided out. A single term inside a sum cannot.
Just for fun

A student "cancels" the 6s in 16/64 and gets 1/4, which happens to be right. Among fractions made of two-digit numbers, this lucky accident only works for 16/64, 19/95, 26/65 and 49/98 (not counting easy ones like 30/60). It even has a name: anomalous cancellation. Do not try it on (x + 6)/(6 + y).

Hands-on activity

Test-a-value detective

You need: a calculator, paper and a pencil.

  1. Choose x = 2. Evaluate (x² − 9)/(x² + 5x + 6) and its simplified form (x − 3)/(x + 2). Do they agree?
  2. Try x = 10 and x = −1 as well.
  3. Now try x = −3 in both. What happens in each?
  4. Use the same method to test a wrong "simplification": (x + 3)/x = 3. Find a value of x that shows it is wrong.

What to look for: x = 2 gives −5/20 = −0.25 and −1/4 = −0.25. x = 10 gives 91/156 = 7/12 ≈ 0.583 for both. x = −1 gives −8/2 = −4 for both. At x = −3 the original gives 0/0 (undefined) while the simplified form gives −6/(−1) = 6: the simplified form only matches the original when the restriction is kept. For (x + 3)/x, almost any x works as a counterexample (every x except 1.5, where (1.5 + 3)/1.5 happens to equal 3), e.g. x = 1 gives 4, not 3.

Your turn

1

Simplify 15a²b/(25ab³) and state the restrictions.

2

Simplify (x² − 4)/(x + 2) and state the restriction.

3

Simplify (3x + 12)/(x² + 4x).

4

Simplify (x² − 5x + 6)/(x² − x − 2).

5

Simplify (x − 7)/(7 − x).

6

State the restrictions on (x + 1)/(x² − 3x − 10).

7

Explain why (x + 5)/5 does not simplify to x.

More practice: warm-up

1

Simplify 8/12.

2

What value is restricted in 4/(x − 9)?

3

Simplify x⁵/x².

4

Factor x² − 49.

5

Simplify 6x/(3x²).

More practice: on your own

1

Simplify (2x² + 5x − 3)/(x² + 6x + 9).

2

Simplify (x³ − 4x)/(x² + 2x).

3

Simplify (9 − x²)/(x² − 2x − 3).

4

Simplify (4x² − 12x + 9)/(4x² − 9).

5

A rectangle has area x² + 7x + 12 and length x + 4. Write and simplify an expression for its width, with any restriction that applies in context.

Challenge

1

Simplify (x² − y²)/(x² − 2xy + y²) and state the restrictions.

2

Simplify (x⁴ − 16)/(x² + 4) and explain why there is no restriction.

3

Write a rational expression that simplifies to (x + 1)/(x − 2) and has restrictions x ≠ 2 and x ≠ 5.

Check your understanding

Quiz

1.

Simplify (x² − 1)/(x² + 3x + 2).

2.

What are the restrictions on (x + 4)/(x² − 16)?

3.

Simplify (3 − x)/(x − 3).

4.

Which simplification is correct?

5.

Why must restrictions be stated from the original expression?

6.

Simplify 20x²y³/(4xy⁵).

For parents and teachers

Big idea: Expectation A3.3 asks students to simplify rational expressions and state the restrictions on the variable. This lesson covers simplifying; the next two cover multiplying, dividing, adding and subtracting.

How to help: Ask your teen to say "factor first" out loud before touching a fraction, and to write restrictions before cancelling. Testing a value (as in the activity) is a great self-check.

Common difficulties:

  • Cancelling terms instead of factors.
  • Stating restrictions from the simplified form and missing the cancelled factor.
  • Missing opposites such as (5 − x) and (x − 5).
  • Factoring errors from Grade 10 (especially ax² + bx + c with a ≠ 1). Revisit the Grade 10 factoring lessons if these are shaky.