--- 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: 2026-11-12 --- > **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 algebraic fractions by factorising both parts, including (x² - 9)/(x² + 7x + 12) - Cancel whole factors only, never single terms inside a sum - State the excluded values that would make the denominator zero ## Lesson: Simplifying algebraic fractions You have factorised expressions and solved zero-factor cases. Use those skills to simplify algebraic fractions. The numerator is the top expression; the denominator is the bottom expression. Factorise both parts, record every input that makes the original denominator zero, then cancel shared whole factors. Keep every original restriction. *(drawing: Record the original exclusions before cancelling.)* Cancellation divides the entire numerator and denominator by the same nonzero factor. In (x² + 7x)/(x + 7), x is a factor of the entire numerator because x² + 7x = x(x + 7). But x is not a factor of the entire denominator x + 7. The shared factor is x + 7. Cancelling it gives x, with x ≠ -7. Factorisation reveals factors that were already present. **Example.** Deleting additive terms is different from dividing whole factors. Deleting x from (x + 8)/(x + 2) leaves 8/2 = 4. Check the allowed input x = 2: the original gives 10/4 = 2.5, not 4. One disagreement at an allowed input disproves equivalence. **Example.** A difference of squares uses x² - a² = (x - a)(x + a). For (4x² - 196)/(4x + 28), the top is 4(x - 7)(x + 7) and the bottom is 4(x + 7). The original bottom excludes -7. Cancel the numerical factor 4 and the shared bracket x + 7. The result is x - 7, with x ≠ -7. For x² + bx + c, find a signed pair with sum b and product c. For ax² + bx + c, split bx using a pair with sum b and product ac, then group. For 2x² + 9x + 4, use 8 and 1: 2x(x + 4) + (x + 4) = (2x + 1)(x + 4). Dividing by x + 4 leaves 2x + 1, with x ≠ -4. **Example.** For (x² + 7x + 6)/(x² - 1), the pair 1 and 6 gives top factors (x + 1)(x + 6). The bottom is (x - 1)(x + 1). Exclude -1 and 1. Cancel x + 1 to get (x + 6)/(x - 1), keeping both exclusions. You must still record the exclusion from the cancelled factor. If no whole factor is shared, nothing cancels. Find exclusions from the original denominator alone. Set each factor equal to zero and solve its case. A bottom x(x - 8) excludes 0 and 8. A repeated factor (x + 9)² excludes -9 once, since exclusions are distinct inputs. A numerator being zero does not exclude an input when the denominator is nonzero. For a requested product, multiply the distinct excluded values with their signs. **Example.** Opposite factors leave a minus sign: 7 - x = -(x - 7). For (x² - 9x + 14)/(7 - x), the top is (x - 2)(x - 7). Exclude x = 7. Rewrite the bottom as -(x - 7), then cancel x - 7. The result is -(x - 2) = 2 - x, with x ≠ 7. **Example.** Factor common numbers and powers of x from entire expressions. For x ≠ 0, dividing 15x and 20x by 5x gives 15x/(20x) = 3/4. A cubic can hide a quadratic factor: x³ - 64x = x(x² - 64). Thus (x³ - 64x)/(x² - 64) becomes x, with x ≠ -8 and x ≠ 8. Both original denominator factors still restrict the input. **Example.** To reconstruct a polynomial numerator, multiply the simplified expression by the original denominator. If the simplified expression is x + 2 and the original denominator is x - 3, the numerator is (x + 2)(x - 3). Expanding gives x² - 3x + 2x - 6 = x² - x - 6. The original fraction has x ≠ 3. **Example.** Cancel one copy of a factor at a time. In (x + 6)/(x + 6)², the denominator contains two copies of x + 6. Only one cancels with the numerator, leaving 1/(x + 6), with x ≠ -6. When the entire numerator cancels, its remaining factor is 1. **Example.** To evaluate, first check the original restrictions, then substitute into the simplified form and divide. The fraction (x² - 36)/(x² + 8x + 12) simplifies to (x - 6)/(x + 2), with x ≠ -6 and -2. Input 3 is allowed. Its value is (3 - 6)/(3 + 2) = -3/5 = -0.6. **Recap.** Factorise the top and the bottom, cancel only whole factors, and state the excluded values from the original denominator. *This lesson, its practice questions and its flashcards were written for this dataset.* ## Practice 30 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) ## Opens up - [Differentiation from First Principles](https://lightmysky.com/learn/mathematics/differentiation-from-first-principles-mt_pdfPztEjw_) - [Rearranging formulae when the subject appears twice](https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj)