MathsICSE Class 9 · Expansions

Cube of a Binomial, a³ ± b³ from Sums and Products, and x ± 1/x Problems (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

The cube identities are (a ± b)³ = a³ ± 3a²b + 3ab² ± b³. They give a³ + b³ = (a + b)³ − 3ab(a + b) and a³ − b³ = (a − b)³ + 3ab(a − b). For x + 1/x = k, x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k; for x − 1/x = k, use k² + 2 and k³ + 3k.

Understand Cube of a Binomial and x ± 1/x Type Problems

Cube of a binomial.
$ $(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} = a^{3} + b^{3} + 3 ab (a + b),$ $
$ $(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3} = a^{3} - b^{3} - 3 ab (a - b) .$ $
The coefficients 1, 3, 3, 1 match Pascal's triangle. For $(2 x - 3 y)^{3}$ take $a = 2 x$ , $b = 3 y$ : $8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3}$ .

Cubes from sums and products. Rearranging:
$ $a^{3} + b^{3} = (a + b)^{3} - 3 ab (a + b), a^{3} - b^{3} = (a - b)^{3} + 3 ab (a - b) .$ $
So if $a + b = 7$ and $ab = 10$ , then $a^{3} + b^{3} = 343 - 210 = 133$ without finding $a$ and $b$ .

** $x + \frac{1}{x}$ problems.** Here $x \cdot \frac{1}{x} = 1$ , so the product term is simply 1.

If $x + \frac{1}{x} = k$ : $x^{2} + \frac{1}{x ^{2}} = k^{2} - 2$ and $x^{3} + \frac{1}{x ^{3}} = k^{3} - 3 k$ .
If $x - \frac{1}{x} = k$ : $x^{2} + \frac{1}{x ^{2}} = k^{2} + 2$ and $x^{3} - \frac{1}{x ^{3}} = k^{3} + 3 k$ .
Relating the two: $(x + \frac{1}{x})^{2} = (x - \frac{1}{x})^{2} + 4$ .

Numbers. $10 1^{3} = (100 + 1)^{3} = 1030301$ and $9 9^{3} = (100 - 1)^{3} = 970299$ .

Check. An identity can be tested with $x = 1$ and $x = 2$ . For $x + \frac{1}{x}$ results, test with a known value such as $x = 2$ ( $k = 2.5$ ).

Level. Class IX is a school-exam class. The chained $x + \frac{1}{x}$ questions below are the hardest school-exam level and a staple of foundation-olympiad practice.

Cube of a Binomial and x ± 1/x Type Problems: formulas to remember

Cube of a sum: $(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$; Coefficients 1, 3, 3, 1.
Cube of a difference: $(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$; Signs alternate.
Sum of cubes from sum: $a^{3} + b^{3} = (a + b)^{3} - 3 ab (a + b)$; No need to find a and b.
x + 1/x: $x^{3} + \frac{1}{x ^{3}} = k^{3} - 3 k, x^{2} + \frac{1}{x ^{2}} = k^{2} - 2$; Where k = x + 1/x.
x − 1/x: $x^{3} - \frac{1}{x ^{3}} = k^{3} + 3 k, x^{2} + \frac{1}{x ^{2}} = k^{2} + 2$; Where k = x − 1/x.

How to solve Cube of a Binomial and x ± 1/x Type Problems questions

Express the target in terms of the given sum (or difference) and product.

1Recognise the pattern: $a + b$ and $ab$ , or $x + 1/ x$ .
2Write the identity that connects the target to the given quantities.
3Substitute and compute step by step (for $x + 1/ x$ , first find the square, then the cube).
4For expansions, write each of the four terms with its sign.
5Test the result with a simple value.

Cube of a Binomial and x ± 1/x Type Problems in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Volume of a cubic container

Suppose a cubic water tank has edge $(2 a - 1)$ metres.

Write its volume in terms of $a$ and find it when $a = 3$ .

Try it first, then show the solutionSolution

$V = (2 a - 1)^{3} = 8 a^{3} - 12 a^{2} + 6 a - 1$ .
At $a = 3$ : $216 - 108 + 18 - 1 = 125$ m³ (edge 5 m, so $5^{3} = 125$ ).

Answer: $8 a^{3} - 12 a^{2} + 6 a - 1$ ; 125 m³.

Memory hook: Expand with a = 2a and b = 1.

Quick mental cube

Imagine finding $9 9^{3}$ without a calculator.

Use an identity.

Try it first, then show the solutionSolution

$9 9^{3} = (100 - 1)^{3} = 1 0^{6} - 3 \cdot 1 0^{4} + 3 \cdot 100 - 1 = 1000000 - 30000 + 300 - 1$ .
$= 970299$ .

Answer: 970299

Memory hook: Round to 100 and correct.

Tricks to remember Cube of a Binomial and x ± 1/x Type Problems

Short memory aids. Say each one aloud once, then use it on a question the same day.

1-3-3-1
Binomial cubes follow the numbers 1, 3, 3, 1.
Use it when: Expansions.
Product is 1
In x and 1/x problems ab = 1, so the 3ab term becomes 3.
Use it when: x + 1/x questions.
Square first
Find x² + 1/x² before x³ + 1/x³ if both are needed.
Use it when: Chained problems.

Cube of a Binomial and x ± 1/x Type Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Coefficient in a cube · 1 mark · MCQBoard level

The coefficient of $x^{2}$ in the expansion of $(x + 2)^{3}$ is:
(a) 2
(b) 4
(c) 6
(d) 8

Try it first, then show the solutionSolution

$(x + 2)^{3} = x^{3} + 6 x^{2} + 12 x + 8$ .

Answer: (c) 6

Example 2: Expanding a cube · 3 marks · Short answerBoard level

Expand $(2 x - 3 y)^{3}$ .

Try it first, then show the solutionSolution

$a = 2 x$ , $b = 3 y$ .
$a^{3} = 8 x^{3}$ ; $3 a^{2} b = 3 (4 x^{2}) (3 y) = 36 x^{2} y$ ; $3 a b^{2} = 3 (2 x) (9 y^{2}) = 54 x y^{2}$ ; $b^{3} = 27 y^{3}$ .
$(2 x - 3 y)^{3} = 8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3}$ .

Answer: $8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3}$

Example 3: x + 1/x = 5 · 4 marks · Short answerHarder board level

If $x + \frac{1}{x} = 5$ , find $x^{2} + \frac{1}{x ^{2}}$ and $x^{3} + \frac{1}{x ^{3}}$ .

Try it first, then show the solutionSolution

$x^{2} + \frac{1}{x ^{2}} = 5^{2} - 2 = 23$ .
$x^{3} + \frac{1}{x ^{3}} = 5^{3} - 3 (5) = 125 - 15 = 110$ .

Answer: $23$ and $110$ .

Example 4: Difference of cubes from difference and product · 4 marks · Short answerHarder board level

If $a - b = 2$ and $ab = 15$ , find $a^{3} - b^{3}$ . Verify by finding $a$ and $b$ (take both positive).

Try it first, then show the solutionSolution

$a^{3} - b^{3} = (a - b)^{3} + 3 ab (a - b) = 8 + 3 (15) (2) = 8 + 90 = 98$ .
From $a - b = 2$ and $ab = 15$ : $(a + b)^{2} = 4 + 60 = 64$ , $a + b = 8$ , so $a = 5$ , $b = 3$ .
Check: $125 - 27 = 98$ . ✓

Answer: $98$ ( $a = 5$ , $b = 3$ ).

Example 5: x − 1/x = 3 · 5 marks · Long answerHOTS / competency

If $x - \frac{1}{x} = 3$ , find $x^{2} + \frac{1}{x ^{2}}$ , $x^{3} - \frac{1}{x ^{3}}$ and $(x + \frac{1}{x})^{2}$ .

Try it first, then show the solutionSolution

$x^{2} + \frac{1}{x ^{2}} = 3^{2} + 2 = 11$ .
$x^{3} - \frac{1}{x ^{3}} = 3^{3} + 3 (3) = 27 + 9 = 36$ .
$(x + \frac{1}{x})^{2} = x^{2} + 2 + \frac{1}{x ^{2}} = 11 + 2 = 13$ (also $(x - \frac{1}{x})^{2} + 4 = 9 + 4 = 13$ ).

Answer: $11$ , $36$ and $13$ .

Where students lose marks in Cube of a Binomial and x ± 1/x Type Problems

The slips examiners see most often, and what to write instead.

Mistake: Wrong signs in (a − b)³
The signs alternate: + − + −.
Wrong working
$(a - b)^{3} = a^{3} - 3 a^{2} b - 3 a b^{2} - b^{3}$ .
Right working
$a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$ .
Do this: Alternate signs.
Usually costs: Whole expansion.
Mistake: Forgetting the 3 in x³ + 1/x³
The formula subtracts 3k, not k.
Wrong working
$x^{3} + \frac{1}{x ^{3}} = 5^{3} - 5 = 120$ .
Right working
$5^{3} - 3 \times 5 = 110$ .
Do this: Write the identity first.
Usually costs: Final answer.
Mistake: Mixing x + 1/x and x − 1/x
The square adds +2 for a difference and −2 for a sum.
Wrong working
$x - \frac{1}{x} = 3 \Rightarrow x^{2} + \frac{1}{x ^{2}} = 9 - 2$ .
Right working
$9 + 2 = 11$ .
Do this: Check the sign inside the bracket.
Usually costs: Final answer.

Exam tips

Class IX is examined in school papers; x ± 1/x chains are common 4 mark questions.
Memorise the four short forms: k² − 2, k³ − 3k, k² + 2, k³ + 3k.
Show the identity you use on its own line.
Verify with the actual numbers where they are easy to find.

Practice questions on Cube of a Binomial and x ± 1/x Type Problems

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The value of $10 1^{3}$ is:
(a) 1030301
(b) 1003001
(c) 1030031
(d) 1303001
Hint
$(100 + 1)^{3}$ .
Answer
(a). $1000000 + 30000 + 300 + 1 = 1030301$ .
Q2 · 3 marks · Short answerHarder board level
Evaluate $9 9^{3}$ using an identity.
Hint
$(100 - 1)^{3}$ .
Answer
$1000000 - 30000 + 300 - 1 = 970299$ .
Q3 · 4 marks · Short answerHarder board level
If $x + y = 7$ and $x y = 10$ , find $x^{3} + y^{3}$ and verify with the actual values.
Hint
$(x + y)^{3} - 3 x y (x + y)$ .
Answer
$343 - 210 = 133$ . Roots are $5$ and $2$ : $125 + 8 = 133$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): If $x + \frac{1}{x} = 3$ , then $x^{3} + \frac{1}{x ^{3}} = 18$ .
Reason (R): $x^{3} + \frac{1}{x ^{3}} = (x + \frac{1}{x})^{3} - 3 (x + \frac{1}{x})$ .
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$27 - 9 = 18$ .
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A cubic box has edge $(2 a - 1)$ cm.
(a) Write the volume as an expression in $a$ .
(b) Find the volume when $a = 3$ .
(c) Find the total surface area when $a = 3$ and say what the edge is.
Hint
At a = 3 the edge is 5.
Answer
(a) $8 a^{3} - 12 a^{2} + 6 a - 1$ (b) $216 - 108 + 18 - 1 = 125$ cm³ (c) the edge is $5$ cm, so the surface area is $6 \times 25 = 150$ cm².

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Expand (a + b)³.Show answer
a³ + 3a²b + 3ab² + b³.
2.a³ + b³ in terms of a + b and ab?Show answer
(a + b)³ − 3ab(a + b).
3.If x + 1/x = k, what is x² + 1/x²?Show answer
k² − 2.
4.If x − 1/x = k, what is x³ − 1/x³?Show answer
k³ + 3k.

Cube of a Binomial and x ± 1/x Type Problems: frequently asked questions

Do I get 1/x problems in the ICSE Class IX syllabus?

Yes. The chapter lists problems of the x + 1/x type under finding values from sums and products.

Why is ab equal to 1 here?

Because x multiplied by 1/x is 1.

Can x be negative?

Yes; the identities hold for any non-zero x.