lesson
Every term meets every term
1 · Learn
To expand two brackets, multiply each term in one bracket by every term in the other, then collect like terms. In (x + 2)(x + 5), an area model shows x², 5x, 2x and 10.
Signs matter in each product. The constant term comes from multiplying the constants, not adding them. Check with substitution after collecting terms.
A third bracket uses the same rule again, not a new shortcut. First collect the product of the first two brackets. Then multiply EVERY term of that expression by EVERY term of the third bracket. For (x² + 7x + 10)(x + 1), multiplying by x gives x³ + 7x² + 10x; multiplying by 1 gives x² + 7x + 10. Add the two rows and collect equal powers.
Powers describe repeated factors: x² × x = x³, while x² + x² = 2x². Likewise a × a × b is a²b, not (ab)². Keep signs attached to terms at every step.
2 · Worked example
(x + 2)(x + 5) = x² + 7x + 10. Also (x - 3)(x + 4) = x² + x - 12. For three brackets: (x + 2)(x + 5)(x + 1) = (x² + 7x + 10)(x + 1) = x³ + 8x² + 17x + 10. At x = 2 the original is 4×7×3 = 84, and the expansion is 8+32+34+10 = 84. Distribution is the reason the identity holds for every x; one numerical check alone is not a proof.
3 · Your turn
First expand (x + 3)(x + 4). Then multiply that expression by (x + 1) and collect. Check both forms at x = 2.
Check your answer
The two brackets give x² + 7x + 12. The three brackets give x³ + 8x² + 19x + 12. At x = 2, both forms of the three-bracket product give 90.
4 · Apply your learning
Draw a four-part area model and verify your expression for x = 2 using both forms.
Adult guidance and safety
Ask for the reasoning and a check, not only a numerical result. Supply accurate diagrams, coordinate grids and data displays where required. Use varied examples and revisit prerequisite gaps; these short teaching models are not a completed assessment programme.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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