--- title: "Simplifying Algebraic Fractions" description: "Factorise the top and the bottom of an algebraic fraction, cancel the common bracket, and state the values of x the fraction cannot take." canonical: https://lightmysky.com/learn/mathematics/simplifying-algebraic-fractions-mt_scDNSoz3eW source: https://lightmysky.com/learn/mathematics/simplifying-algebraic-fractions-mt_scDNSoz3eW.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Simplifying Algebraic Fractions Factorise the top and the bottom of an algebraic fraction, cancel the common bracket, and state the values of x the fraction cannot take. Subject: Mathematics · Area: Algebra · Ages 14 to 15 Page: https://lightmysky.com/learn/mathematics/simplifying-algebraic-fractions-mt_scDNSoz3eW ## Ready when they can - Simplify (x² - 9)/(x² + 7x + 12) by factorising both parts - Cancel whole factors only, never single terms inside a sum - State the excluded values that would make the denominator zero ## Lesson: Simplifying Algebraic Fractions The last four stops gave you a factorising toolkit: the pair method, the special patterns, splitting the middle term, and solving equations from brackets. This stop applies it to fractions. An algebraic fraction has expressions in the top and the bottom. Simplifying means factorising the top and the bottom, cancelling any whole factor they share, and stating the values of x the fraction cannot take. *(drawing: The four moves, in the order they happen.)* Only whole factors cancel. In (x squared + 3x)/(x + 3), the x in the top is a term in a sum, not a factor, so it cannot be cancelled against the x in the bottom. The slip is to cancel the x's and the 3's and write x + 3. The top is x(x + 3), so the whole bracket (x + 3) cancels and the answer is x. When in doubt, factorise first: if a factor is not visible, it is not there. **Example.** Simplify (x squared - 4)/(x + 2). Factorise the top as a difference of two squares: (x - 2)(x + 2). The bracket (x + 2) appears in the bottom, so cancel it and the fraction becomes x - 2. The cancelled bracket came from the original denominator, so x = -2 is still not allowed: the original fraction has a zero bottom there. The simplified form x - 2 is fine at -2, but that does not repair the original. **Example.** Simplify (2x squared + 5x - 3)/(x + 3). Factorise the top by splitting the middle term: the product is -6, and the pair that adds to 5 is 6 and -1, so the top is (2x - 1)(x + 3). The bracket (x + 3) cancels with the bottom, leaving 2x - 1. State the excluded value: x = -3 makes the original denominator zero, so the fraction cannot take it. **Example.** Now factorise both parts. Simplify (x squared + 5x + 4)/(x squared - 1). The top is (x + 1)(x + 4), using the pair 1 and 4. The bottom is a difference of two squares: (x - 1)(x + 1). Cancel the shared bracket (x + 1) to get (x + 4)/(x - 1). Both brackets in the original denominator exclude a value, so x = -1 and x = 1 are both not allowed. **Tip.** The excluded values always come from the original denominator, before you cancel anything. A bracket that cancels still forbids the value that makes it zero, because the original fraction was undefined there. Write them down every time; they are part of the answer. **Recap.** Factorise the top and the bottom, cancel only whole factors, and state the excluded values from the original denominator. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Solving Quadratic Equations by Factorising](https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU) - [Dividing fractions](https://lightmysky.com/learn/mathematics/dividing-fractions-mt_9Y96vxG_LH) ## Opens up - [Rearranging Formulae When the Subject Appears Twice](https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj) - [Differentiation from First Principles](https://lightmysky.com/learn/mathematics/differentiation-from-first-principles-mt_pdfPztEjw_)