medium difficulty6 min

Simplifying Rational Expressions

Simplifying rational expressions means factorising numerator and denominator, cancelling common non-zero factors, and stating restrictions where needed.

In one line: Cancel factors, never pieces of sums.

Practice This Concept

Understand it

In algebraic fractions, cancellation is allowed only for common factors, not for separate terms. Identities help convert expressions into factor form, such as x²-9=(x-3)(x+3). A value that makes the original denominator zero must be excluded.

Example

(x²-9)/(x-3) = ((x-3)(x+3))/(x-3) = x+3, where x cannot be 3.

Formula and rule

Use identities like a²-b²=(a-b)(a+b) and perfect-square trinomials before cancellation; denominator factors cannot be zero.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Simplify (x²+6x+9)/(x+3).

1The numerator is (x+3)².
2Cancelling one common factor gives x+3, with x not equal to -3.

Method you can reuse

Factorise numerator and denominator, identify common full factors, cancel them, write the simplified expression, and record values that make the original denominator zero.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is (x²-16)/(x-4) simplified as x+4 with x not equal to 4?

Think first, then show answer

Because x²-16 factorises as (x-4)(x+4), so the common factor x-4 cancels, but x=4 is still not allowed in the original denominator.

Common wrong answer

Simplifying (x²+5x)/(x) as x²+5 is wrong; factor first to get x(x+5)/x, so the result is x+5 with x not equal to 0.

Don't mix this up

Students cancel terms across addition or subtraction, such as cancelling x in (x+2)/x, which is not valid.

Exam tip

Factor first, cancel only full common factors, and mention the denominator restriction if the expression has variables.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions often ask simplification of algebraic fractions and may expect the excluded value of the variable.

1
Enough for 1 mark
Simplifying rational expressions means factorising numerator and denominator, cancelling common non-zero factors, and stating restrictions where needed.
2
Add for 2 marks
Example: (x²-9)/(x-3) = ((x-3)(x+3))/(x-3) = x+3, where x cannot be 3.
3
Add for 3 marks
In algebraic fractions, cancellation is allowed only for common factors, not for separate terms.

MCQ Quiz

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