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225 lines
9.2 KiB
HTML
225 lines
9.2 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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Long before modern symbols, people solved problems that required multiplying lengths and
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collecting areas — the same ideas behind expanding and factorising. Babylonian scribes (around
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1800 BCE) recorded rectangle problems on clay tablets: given area and side relationships,
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find unknown lengths. Their recipes are early cousins of “expand, then simplify.”
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</p>
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<figure class="book-figure">
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<img
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class="book-figure__img"
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src="/chapters/math/polynomials-expansion-factorisation/polynomials-history.png"
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alt="Illustrated timeline: Babylonian area on clay tablets, al-Khwarizmi's algebra, Viète's difference of squares, Enlightenment science, and modern expand–factorise steps"
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width="960"
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height="640"
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loading="lazy"
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decoding="async"
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/>
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<figcaption class="book-figure__caption">
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From geometric area puzzles to symbolic algebra: how expansion and factorisation became
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standard tools in school mathematics.
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</figcaption>
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</figure>
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<p>
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In the 9th century, <strong>al-Khwarizmi</strong> organised methods for equations and products
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in Arabic texts that later spread to Europe. Renaissance mathematicians such as
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<strong>Viète</strong> introduced letters for unknowns and constants, making it easier to see
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patterns like \(a^2 - b^2\). By the 17th–18th centuries, polynomial algebra was central to
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calculus and science; factorisation became a routine step before solving equations or
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simplifying formulas.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation</h2>
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<h3 id="area-distributive" class="book-prose__heading">Area proof of \((a+b)(c+d)\)</h3>
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<p>
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Draw a rectangle with width \(a+b\) and height \(c+d\). The total area is \((a+b)(c+d)\). Split
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the rectangle into four smaller rectangles with areas \(ac\), \(ad\), \(bc\), and \(bd\). The
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sum of partial areas equals the whole:
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\[ (a+b)(c+d) = ac + ad + bc + bd. \]
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This is the distributive law in two dimensions — the same picture as in the area-model widget
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above.
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</p>
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<h3 id="derive-identities" class="book-prose__heading">Deriving standard identities</h3>
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<p>
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Set \(c = a\) and \(d = b\) in the area formula for a square of side \(a+b\):
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\[ (a+b)^2 = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2. \]
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For a square of side \(a-b\) (with \(a > b\)), remove a strip of width \(b\) from a square of
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side \(a\); the remaining L-shape rearranges to show
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\[ (a-b)^2 = a^2 - 2ab + b^2. \]
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Difference of squares: a large square of side \(a\) minus a small square of side \(b\) leaves two
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congruent rectangles, which rearrange to a rectangle of dimensions \((a+b)\) by \((a-b)\):
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\[ a^2 - b^2 = (a+b)(a-b). \]
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</p>
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<h3 id="grouping-walkthrough" class="book-prose__heading">Grouping walkthrough</h3>
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<p>
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Factorise \(3x^2 + 12x - 2x - 8\).
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</p>
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<ol>
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<li>Group: \((3x^2 + 12x) + (-2x - 8)\).</li>
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<li>Factor each pair: \(3x(x+4) - 2(x+4)\).</li>
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<li>Common bracket \((x+4)\): \((3x-2)(x+4)\).</li>
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</ol>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<p>
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Pause and reason before continuing. Discuss with a partner or write your work in a notebook.
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</p>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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In \(4x^3 - 2x^2 + 5\), what is the degree? Which terms are like terms with \(4x^3\)?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Expand \((2x-1)(x+4)\) without skipping steps. How many terms do you get before collecting?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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How would you factorise \(x^2 - 16\)? Which identity applies?
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</p>
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</div>
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<div class="book-callout" role="note">
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<p class="book-callout__label">Checkpoint</p>
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<p>
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Expand \(x(x+1) - (x+1)\). What common factor appears? What is the factorised form?
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</p>
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</div>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<h3 id="geometry-area" class="book-prose__heading">Geometry: area and perimeter</h3>
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<p>
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A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Its area is
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\[ (x+3)(2x-1) = 2x^2 + 5x - 3 \quad\text{cm}^2. \]
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The perimeter is \(2\bigl((x+3)+(2x-1)\bigr) = 2(3x+2) = 6x+4\) cm. Expansion links algebraic
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descriptions of sides to measurable quantities.
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</p>
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<h3 id="revenue-profit" class="book-prose__heading">Revenue and pricing</h3>
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<p>
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If a shop sells \(n\) items at price \((50 - 0.5n)\) dollars each, revenue is
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\(n(50 - 0.5n) = 50n - 0.5n^2\). Expanding a product of two linear factors often appears before
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finding maximum revenue (a quadratic in \(n\)).
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</p>
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<h3 id="number-pattern" class="book-prose__heading">Number patterns</h3>
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<p>
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Notice that \(11^2 = 121\), \(111^2 = 12321\). In general, expanding \((10^k + 10^{k-1} +
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\cdots + 1)^2\) explains palindromic squares — a playful link between identities and arithmetic
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patterns.
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</p>
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<h2 id="question-bank" class="book-prose__heading">Question bank</h2>
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<p>
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Work each question, then open the solution to check your reasoning. Difficulty increases
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through the sets.
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</p>
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<h3 id="questions-easy" class="book-prose__heading">Easy</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Expand \(3(x+2)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(3x + 6\).</p>
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<p><strong>Steps:</strong> \(3 \cdot x + 3 \cdot 2 = 3x + 6\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Simplify \(2x + 5x - 1\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(7x - 1\).</p>
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<p><strong>Explanation:</strong> Combine like terms \(2x + 5x = 7x\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Factorise \(6x + 9\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(3(2x + 3)\).</p>
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<p><strong>Steps:</strong> HCF is \(3\); \(6x \div 3 = 2x\), \(9 \div 3 = 3\).</p>
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</div>
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</details>
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<h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Expand \((x+4)(x-2)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(x^2 + 2x - 8\).</p>
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<p><strong>Steps:</strong> \(x^2 - 2x + 4x - 8 = x^2 + 2x - 8\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Factorise \(x^2 - 9\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((x+3)(x-3)\).</p>
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<p><strong>Explanation:</strong> Difference of squares: \(x^2 - 3^2\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Factorise \(x^2 + 6x + 9\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((x+3)^2\).</p>
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<p><strong>Explanation:</strong> Perfect square: \(3 \times 3 = 9\), \(3 + 3 = 6\).</p>
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</div>
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</details>
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<h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Factorise \(2x^2 + 6x - x - 3\) by grouping.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((2x-1)(x+3)\).</p>
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<p><strong>Steps:</strong> \(2x(x+3) - 1(x+3) = (2x-1)(x+3)\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Factorise \(2x^2 + 7x + 3\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((2x+1)(x+3)\).</p>
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<p><strong>Steps:</strong> \(ac = 6\); split \(7x = 6x + x\); \(2x(x+3) + 1(x+3) = (2x+1)(x+3)\).</p>
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</div>
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</details>
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<h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>
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<details class="book-question">
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<summary class="book-question__prompt">Expand and then factorise \(x(x+2) - (x+2)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((x+2)(x-1)\) after factorising (expansion gives \(x^2 + x - 2\)).</p>
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<p><strong>Steps:</strong> Factor common \((x+2)\): \((x+2)(x-1)\). Check: \((x+2)(x-1) = x^2 + x - 2\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A rectangle has sides \((x+3)\) cm and \((2x-1)\) cm. Find its area in expanded form.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \((2x^2 + 5x - 3)\) cm².</p>
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<p><strong>Calculation:</strong> \((x+3)(2x-1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3\).</p>
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</div>
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</details>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>al-Khwarizmi, <em>Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala</em> (c. 820 CE) — systematic treatment of algebra.</li>
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<li>HKDSE Compulsory Part curriculum — Number and Algebra strand (polynomials, expansion, factorisation).</li>
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</ul>
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</div>
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