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Simplifying Rational Expressions

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

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Student-friendly explanation

In algebraic fractions, cancellation is allowed only for common factors, not for separate terms. Identities help convert expressions into factor form, such as x^2-9=(x-3)(x+3). A value that makes the original denominator zero must be excluded.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    (x^2-9)/(x-3) = ((x-3)(x+3))/(x-3) = x+3, where x cannot be 3.

  4. 4

    Avoid this incomplete answer

    Simplifying (x^2+5x)/(x) as x^2+5 is wrong; factor first to get x(x+5)/x, so the result is x+5 with x not equal to 0.

Definition

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

Example

(x^2-9)/(x-3) = ((x-3)(x+3))/(x-3) = x+3, where x cannot be 3.

Rule to remember

Use identities like a^2-b^2=(a-b)(a+b) and perfect-square trinomials before cancellation; denominator factors cannot be zero.

Memory hook

Cancel factors, never pieces of sums.

Examples and method

Worked example

Simplify (x^2+6x+9)/(x+3). The numerator is (x+3)^2. Cancelling one common factor gives x+3, with x not equal to -3.

Method to apply

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Simplifying (x^2+5x)/(x) as x^2+5 is wrong; factor first to get x(x+5)/x, so the result is x+5 with x not equal to 0.

Exam tip

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

Quick check

Why is (x^2-16)/(x-4) simplified as x+4 with x not equal to 4?

Because x^2-16 factorises as (x-4)(x+4), so the common factor x-4 cancels, but x=4 is still not allowed in the original denominator.

Answer writing and exam use

1-mark answer

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

2-mark answer

Simplifying rational expressions means factorising numerator and denominator, cancelling common non-zero factors, and stating restrictions where needed. Use identities like a^2-b^2=(a-b)(a+b) and perfect-square trinomials before cancellation; denominator factors cannot be zero. (x^2-9)/(x-3) = ((x-3)(x+3))/(x-3) = x+3, where x cannot be 3.

3-mark answer

In algebraic fractions, cancellation is allowed only for common factors, not for separate terms. Identities help convert expressions into factor form, such as x^2-9=(x-3)(x+3). A value that makes the original denominator zero must be excluded. Use identities like a^2-b^2=(a-b)(a+b) and perfect-square trinomials before cancellation; denominator factors cannot be zero. Simplify (x^2+6x+9)/(x+3). The numerator is (x+3)^2. Cancelling one common factor gives x+3, with x not equal to -3. Questions often ask simplification of algebraic fractions and may expect the excluded value of the variable. Simplifying (x^2+5x)/(x) as x^2+5 is wrong; factor first to get x(x+5)/x, so the result is x+5 with x not equal to 0.
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