CDS Mathematics · Algebraic Identities and Simplification

Squares and Cubes of a Binomial

Knowing a + b and ab fixes every symmetric expression in a and b, and the standard factor forms turn long arithmetic into a single subtraction.

All Algebraic Identities and Simplification notes CDS Mathematics strategy

Why this matters

Twelve PYQs. The core move is to write a² + b² or a³ + b³ through a + b and ab instead of finding a and b. The other two moves are spotting an expanded cube and using the factor forms of a² − b² and a³ ± b³ on awkward numbers.

Concept 1 of 3: Everything from a + b and ab

Any expression that does not change when

a

and

b

swap can be rebuilt from their sum and product. So when a question gives

a + b

and

ab

, never solve for

a

and

b

: expand a power of the sum and remove what you do not want.

Definition

$a^2 + b^2 = (a + b)^2 - 2ab$
$a^3 + b^3 = (a + b)^3 - 3ab(a + b)$
$a^3 - b^3 = (a - b)^3 + 3ab(a - b)$
$(a - b)^2 = (a + b)^2 - 4ab$ , and $\dfrac1a + \dfrac1b = \dfrac{a + b}{ab}$ .

When square roots appear, as in $a\sqrt a + b\sqrt b$ , put $p = \sqrt a$ and $q = \sqrt b$ so the expression becomes $p^3 + q^3$ .

Cube of a sum

a^3 + b^3 = (a + b)^3 - 3ab(a + b)

Worked example

a + b = 7

and

ab = 10

, find

a^2 + b^2

and

a^3 + b^3

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q33Easy

Example 1 · Algebraic Identities and Simplification · Squares and Cubes of a Binomial

a + b = 5

and

ab = 6

, then what is the value of

a^3 + b^3

The correction term is 3ab(a + b)

(a + b)^3

a^3 + b^3 + 3ab(a + b)

, not

a^3 + b^3 + 3ab

. Forgetting the factor

(a + b)

gives a wrong answer that is often printed as an option.

Drill 5 more on everything from a + b and ab

Concept 2 of 3: Spotting an expanded cube

Four terms with coefficients in the pattern

1, 3, 3, 1

(after taking out the cubes) are a cube that has been expanded. Collapse it back and the given value does the rest.

Definition

$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ and $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ .
Test: are the first and last terms perfect cubes $a^3$ and $b^3$ ? Is the second term $3a^2b$ ? If so, the four terms are $(a \pm b)^3$ .
The same test works with fractions: $\dfrac{1}{t^3} - \dfrac{6}{t^2} + \dfrac{12}{t} - 8 = \left(\dfrac1t - 2\right)^3$ .

Cube of a difference

(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Worked example

x - 2y = 3

, find

x^3 - 6x^2y + 12xy^2 - 8y^3 + 5

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q40Moderate

Example 2 · Algebraic Identities and Simplification · Squares and Cubes of a Binomial

2x-3y-7=0

, then what is the value of

8x^3-36x^2y+54xy^2-27y^3-340

Check the middle coefficients

27x^3 + 54x^2y + 36xy^2 + 8y^3

(3x + 2y)^3

because

3(3x)^2(2y) = 54x^2y

. If a middle term does not match, the four terms are not a cube and the shortcut fails.

Drill 1 more on spotting an expanded cube

Concept 3 of 3: Difference of squares, sum and difference of cubes

The factor forms turn a hard-looking numerical fraction into one subtraction. When the denominator is

a^2 \pm ab + b^2

, look for

a^3 \pm b^3

on top.

Definition

$a^2 - b^2 = (a - b)(a + b)$
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
$(p + q)^2 + (p - q)^2 = 2(p^2 + q^2)$ and $(p + q)^2 - (p - q)^2 = 4pq$
$x^4 - 1 = (x - 1)(x + 1)(x^2 + 1)$ .

Sum and difference of cubes

a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)

Worked example

Evaluate

\dfrac{7.3^3 + 2.7^3}{7.3^2 - 7.3\times 2.7 + 2.7^2}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — Elementary Mathematics · Q6Moderate

Example 3 · Algebraic Identities and Simplification · Squares and Cubes of a Binomial

What is the value of the following?

\frac{(5.4)^3 - 0.064}{(5.4)^2 + 2.16 + 0.16}

The signs pair up opposite

a^3 + b^3

goes with

a^2 - ab + b^2

, and

a^3 - b^3

with

a^2 + ab + b^2

. Read the sign of the middle term in the denominator first; it tells you which cube to look for on top.

Drill 3 more on difference of squares, sum and difference of cubes

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Everything from a + b and ab
Cube of a sum
$a^3 + b^3 = (a + b)^3 - 3ab(a + b)$
Spotting an expanded cube
Cube of a difference
$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
Difference of squares, sum and difference of cubes
Sum and difference of cubes
$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

Watch out for (3)

The correction term is 3ab(a + b)
→ Everything from a + b and ab
Check the middle coefficients
→ Spotting an expanded cube
The signs pair up opposite
→ Difference of squares, sum and difference of cubes

Test yourself on Algebraic Identities and Simplification

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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