This course turns the distributive law into a set of ready-made results. It starts with the eight forms of the distributive law and the product of a sum and a difference. It then proves the identities for the square of a sum and of a difference, the results got by adding and subtracting these squares, and the square of a trinomial with any signs. The last two lessons cover the cube of a binomial, the sum and difference of two cubes, and the identity for a³ + b³ + c³ − 3abc. Each identity is proved and then used to expand, to calculate quickly and to find values without solving.
Middle Years
Algebraic Identities
The distributive laws and the standard algebraic identities for squares and cubes of binomials and trinomials, with proofs, expansions and finding values.
- 6 lessons
Overview
Lessons
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Distributive laws and the difference of squares
The eight forms of the distributive law, what an identity is, and the product (a + b)(a − b) = a² − b² used for expressions and for quick arithmetic.
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Square of a binomial
Proofs of (a + b)² and (a − b)², the missing middle term, and expanding squares with whole terms, fractions and numbers.
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Sums and differences of squares
Adding and subtracting (a + b)² and (a − b)², rearranged forms, showing two expressions are equal, and finding values such as x² + 1/x².
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Square of a trinomial
The identity for (a + b + c)², how the signs change when terms are negative, expansions, and finding sums of squares from sums and products.
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Cube of a binomial
Proofs of (a + b)³ and (a − b)³ in long and short form, expanding cubes, cubing numbers quickly, and finding values such as x³ + 1/x³.
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Sum and difference of cubes
Forms of a³ + b³ and a³ − b³, the identity for a³ + b³ + c³ − 3abc, and why a + b + c = 0 makes the sum of cubes equal to 3abc.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Algebraic Identities
Typed from the teacher's handwritten class slides, 7 pages.
Download the notes (PDF, 324.5 KB)