SPI Mathematics · Free notes

Algebraic Identities and Simplification: SPI Mathematics

Algebraic Identities and Simplification has 16 questions in the SPI bank, and two of them are from the 2022 paper. Ten are EASY, six are MODERATE and none is HARD. Every one is decided by recognising which identity the expression is: once you see a perfect square, a difference of squares or a + b + c = 0, the answer is a line or two. So learn the identities until you can read them backwards, from the expanded form to the factors, as fast as forwards.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

SPI home All formulas on one page

Subtopic notes

Formula & revision sheet

7 formulas · 12 gotchas across all subtopics — the exam-eve cheat-sheet

Factorising with Common Factors and Identities

Formulas (4)

Taking out a common factor and factorising by grouping · Common factor $ab + ac = a(b + c)$
The identity for (x + a)(x + b) · Product of two binomials $(x + a)(x + b) = x^2 + (a + b)x + ab$
Perfect square identities · Perfect squares $(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2$
Difference of two squares · Difference of squares $a^2 - b^2 = (a + b)(a - b) \qquad x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2$

Watch out for (7)

Giving the common bracket alone
→ Taking out a common factor and factorising by grouping
Pairing terms that share nothing
→ Taking out a common factor and factorising by grouping
Getting the middle sign wrong
→ The identity for (x + a)(x + b)
Dropping the common factor
→ Perfect square identities
Forgetting the middle term
→ Perfect square identities
Factorising a sum of squares
→ Difference of two squares
Calling a hidden difference 'not possible'
→ Difference of two squares

Finding Values with Identities

Formulas (3)

Squares and cubes from the sum and the product · Sum and product identities $x^2 + y^2 = (x + y)^2 - 2xy \qquad x^3 + y^3 = (x + y)^3 - 3xy(x + y) \qquad x^3 - y^3 = (x - y)^3 + 3xy(x - y)$
Three-term identities and the condition a + b + c = 0 · Three-term identities $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \qquad a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc$
Simplifying a chain of brackets that cancel · One bracket as one fraction $1 + \frac{1}{n} = \frac{n + 1}{n} \qquad 1 - \frac{1}{n} = \frac{n - 1}{n}$

Watch out for (5)

Squaring the sum and stopping
→ Squares and cubes from the sum and the product
Leaving out the bracket in the cube identity
→ Squares and cubes from the sum and the product
Forgetting the 2 in the square of three terms
→ Three-term identities and the condition a + b + c = 0
Dropping the signs of the three terms
→ Three-term identities and the condition a + b + c = 0
Multiplying the brackets out
→ Simplifying a chain of brackets that cancel

Question weightage by concept

7 concepts · 16 questions — where the marks actually sit, so you know what to drill first

Factorising with Common Factors and Identities11 questions · 69%

Concept	Qs	Share
Taking out a common factor and factorising by grouping	3	19%
The identity for (x + a)(x + b)	3	19%
Difference of two squares	3	19%
Perfect square identities	2	13%

Finding Values with Identities5 questions · 31%

Concept	Qs	Share
Squares and cubes from the sum and the product	2	13%
Three-term identities and the condition a + b + c = 0	2	13%
Simplifying a chain of brackets that cancel	1	6%

Test yourself on a real paper

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