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ICSE • कक्षा 8 • Mathematics • अध्याय 13
अनुमानित समय: 45 Mins
प्रगति: अध्ययनरत

बीजगणितीय सर्वसमिकाएँ (Algebraic Identities)

In ICSE Class 8 Mathematics, "Algebraic Identities" provides an authoritative, mathematically rigorous master study guide investigating standard polynomial identities, geometric visualizations, expansions, conditional identities, and numerical evaluations. This comprehensive chapter explores What is an Identity? (Equality valid for all possible numerical values of variables vs a conditional equation valid only for specific roots), Standard Linear and Quadratic Identities: 1. $(a + b)^2 = a^2 + 2ab + b^2$, 2. $(a - b)^2 = a^2 - 2ab + b^2$, 3. $(a + b)(a - b) = a^2 - b^2$, 4. $(x + a)(x + b) = x^2 + (a + b)x + ab$, Derived Coupled Identities ($(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$; $(a + b)^2 - (a - b)^2 = 4ab$), Reciprocal Identical Expansions ($(x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2$; $(x - \frac{1}{x})^2 = x^2 + \frac{1}{x^2} - 2$; $(x + \frac{1}{x})^2 - (x - \frac{1}{x})^2 = 4$), Trinomial Expansion ($(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$; Handling negative signs by substitution), Cubic Expansions: 1. $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$, 2. $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$, Sum and Difference of Two Cubes ($a^3 + b^3 = (a + b)(a^2 - ab + b^2)$; $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$), Conditional Cubic Identity (If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$), and Rapid Numerical Mental Evaluation without long multiplication (e.g., $105^2 = (100 + 5)^2, 98 \times 102 = (100 - 2)(100 + 2)$) aligned with the 2026–27 CISCE ICSE curriculum.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें
प्रस्तावना अध्याय अवधारणाएँ सूत्र एवं सर्वसमिकाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

How Did an Indian Schoolboy Calculate 998 Squared in Two Seconds in His Head While His Teacher Was Still Writing the First Digit?

Imagine someone asking you to multiply $998 \times 998$. Most people grab a pen and start laboriously writing three rows of long multiplication, adding carries and sweating over errors. But a student who knows algebraic identities smiles, closes their eyes, and says: "996,004!" How did they do it in two seconds? They didn't multiply $998$ by $998$; they saw $998$ as $(1,000 - 2)$! Using the identity $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 1,000$ and $b = 2$: $1,000^2 - 2(1000)(2) + 2^2 = 1,000,000 - 4,000 + 4 = \mathbf{996,004}$! Clean, instant mental arithmetic! An Algebraic Identity is an equality that holds true for EVERY NUMBER in the universe! What is the secret connection between $(x + \frac{1}{x})$ and $(x^2 + \frac{1}{x^2})$? Why does $28^3 + (-15)^3 + (-13)^3$ collapse into a simple product of three numbers without calculating any cubes? Let's master algebraic identities.

यह अध्याय क्यों महत्वपूर्ण है

Algebraic identities are the primary algebraic tool across geometry (distance formulas, coordinate rotation), calculus (derivatives of powers, binomial theorem), quantum mechanics, computer graphics ray tracing, and engineering stress tensors. Mastering expansions and conditional proofs is a core requirement in ICSE mathematics.

अध्ययन से पूर्व (आवश्यक ज्ञान)

  • Multiplication of polynomials and exponents from Chapters 3 and 12.
  • Factorisation fundamentals from Chapter 1.
  • Order of operations.

इस अध्याय के लक्ष्य

  • Differentiate between an algebraic identity and a conditional equation.
  • Apply the standard quadratic identities $(a \pm b)^2$ and $(a + b)(a - b)$ for expansion and mental arithmetic.
  • Expand three-term polynomials using $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.
  • Expand cubic expressions using $(a \pm b)^3 = a^3 \pm b^3 \pm 3ab(a \pm b)$.
  • Utilize reciprocal identities involving $(x + 1/x)$ and $(x^2 + 1/x^2)$.
  • Apply the conditional identity: If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.

अध्याय रूपरेखा एवं प्रगति

1 1. Identity vs Equation & Standard...
2 2. Reciprocal Identities: $(x \pm \...
3 3. Trinomial Square & Cubic Expansi...
4 4. Sum/Difference of Cubes & The Co...

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. Identity vs Equation & Standard Quadratic Identities

Understand
A. Identity vs Equation:
  • An Equation is an algebraic equality that is true only for specific numerical roots (e.g., $2x + 3 = 7$ is true only for $x = 2$).
  • An Identity is an algebraic equality that is true for ALL possible numerical values assigned to its variables (e.g., $(x + 1)^2 = x^2 + 2x + 1$ is true for every real number).
B. The Standard Quadratic Identities:
  1. $\mathbf{(a + b)^2 = a^2 + 2ab + b^2}$
  2. $\mathbf{(a - b)^2 = a^2 - 2ab + b^2}$
  3. $\mathbf{(a + b)(a - b) = a^2 - b^2}$
  4. $\mathbf{(x + a)(x + b) = x^2 + (a + b)x + ab}$
C. Coupled Auxiliary Identities:
$$\mathbf{(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)}$$ $$\mathbf{(a + b)^2 - (a - b)^2 = 4ab}$$

2. Reciprocal Identities: $(x \pm \frac{1}{x})$

Reciprocal Identities

When $b = \frac{1}{x}$, the cross-term $2ab = 2(x)(\frac{1}{x}) = 2$ becomes a pure constant!

$$\mathbf{\left( x + \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} + 2 \implies x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2}$$ $$\mathbf{\left( x - \frac{1}{x} \right)^2 = x^2 + \frac{1}{x^2} - 2 \implies x^2 + \frac{1}{x^2} = \left( x - \frac{1}{x} \right)^2 + 2}$$ $$\mathbf{\left( x + \frac{1}{x} \right)^2 - \left( x - \frac{1}{x} \right)^2 = 4}$$

3. Trinomial Square & Cubic Expansions

Trinomial & Cubic
A. Square of a Trinomial:
$$\mathbf{(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca}$$

Handling Negative Signs: Treat $(a - b - c)^2$ as $[a + (-b) + (-c)]^2$:

$$(a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca$$
B. Expansion of Cubes:
  1. $\mathbf{(a + b)^3 = a^3 + b^3 + 3ab(a + b) = a^3 + 3a^2b + 3ab^2 + b^3}$
  2. $\mathbf{(a - b)^3 = a^3 - b^3 - 3ab(a - b) = a^3 - 3a^2b + 3ab^2 - b^3}$
  3. Reciprocal Cubes: $$\left( x + \frac{1}{x} \right)^3 = x^3 + \frac{1}{x^3} + 3\left( x + \frac{1}{x} \right)$$ $$\left( x - \frac{1}{x} \right)^3 = x^3 - \frac{1}{x^3} - 3\left( x - \frac{1}{x} \right)$$

4. Sum/Difference of Cubes & The Conditional Identity

Conditional Identities
A. Factorisation Identities for Cubes:
$$\mathbf{a^3 + b^3 = (a + b)(a^2 - ab + b^2)}$$ $$\mathbf{a^3 - b^3 = (a - b)(a^2 + ab + b^2)}$$
B. The Famous Conditional Identity:

The general three-cube identity states:

$$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$$ $$\mathbf{\text{If } a + b + c = 0, \quad \text{then } a^3 + b^3 + c^3 = 3abc}$$

Example: Evaluate $28^3 + (-15)^3 + (-13)^3$:
Check sum: $a + b + c = 28 + (-15) + (-13) = 28 - 28 = 0$.
Therefore: $28^3 + (-15)^3 + (-13)^3 = 3(28)(-15)(-13) = \mathbf{16,380}$!

महत्वपूर्ण सूत्र, सर्वसमिकाएँ एवं प्रमेय

Trinomial Expansion Formula
$$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$
Square of a three-term polynomial.
Conditional Cubic Identity
$$a + b + c = 0 \implies a^3 + b^3 + c^3 = 3abc$$
Permits instant evaluation without computing individual cubes.

Algebra: Geometric Proof of (a+b)^2 & Cubic Identities

Algebraic Identities: Geometric Visualizations & Cubic Expansions GEOMETRIC PROOF OF (a + b)^2 a2 ab ab b2 Area = a2 + ab + ab + b2 = a2 + 2ab + b2 CUBIC & CONDITIONAL IDENTITIES • Cubic Expansion: (a + b)3 = a3 + b3 + 3ab(a + b) (a - b)3 = a3 - b3 - 3ab(a - b) • Trinomial Square: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca THE CONDITIONAL THEOREM: If a + b + c = 0 ⇒ a3 + b3 + c3 = 3abc • Reciprocals: (x + 1/x)2 = x2 + 1/x2 + 2 (a+b)^2 = a^2+2ab+b^2 • (a+b)^3 = a^3+b^3+3ab(a+b) • a+b+c=0 ⇒ a^3+b^3+c^3 = 3abc

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1
An algebraic identity is an equality true for all values of its variables.
मुख्य बिंदु 2
Standard square identities: (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2.
मुख्य बिंदु 3
Difference of squares: (a + b)(a - b) = a^2 - b^2.
मुख्य बिंदु 4
Coupled auxiliary formulas: (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) and (a + b)^2 - (a - b)^2 = 4ab.
मुख्य बिंदु 5
Reciprocal identities: (x + 1/x)^2 = x^2 + 1/x^2 + 2 and (x - 1/x)^2 = x^2 + 1/x^2 - 2.
मुख्य बिंदु 6
Trinomial expansion: (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
मुख्य बिंदु 7
Cubic expansions: (a + b)^3 = a^3 + b^3 + 3ab(a + b) and (a - b)^3 = a^3 - b^3 - 3ab(a - b).
मुख्य बिंदु 8
Sum and difference of cubes: a^3 +- b^3 = (a +- b)(a^2 -+ ab + b^2).
मुख्य बिंदु 9
Conditional identity: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.
मुख्य बिंदु 10
Identities enable rapid mental arithmetic evaluations without long multiplication.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

1
If $x + \frac{1}{x} = 5$, find the values of: (a) $x^2 + \frac{1}{x^2}$, (b) $x^4 + \frac{1}{x^4}$.
उत्तर एवं व्याख्या देखें
उत्तर:

• (a) Find $x^2 + \frac{1}{x^2}$:
Square both sides of $x + \frac{1}{x} = 5$:

$$\left( x + \frac{1}{x} \right)^2 = 5^2$$


$$x^2 + 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = 25$$


$$x^2 + 2 + \frac{1}{x^2} = 25$$


$$x^2 + \frac{1}{x^2} = 25 - 2 = \mathbf{23}$$



• (b) Find $x^4 + \frac{1}{x^4}$:
Square both sides of $x^2 + \frac{1}{x^2} = 23$:

$$\left( x^2 + \frac{1}{x^2} \right)^2 = 23^2$$


$$x^4 + 2 + \frac{1}{x^4} = 529$$


$$x^4 + \frac{1}{x^4} = 529 - 2 = \mathbf{527}$$

.


(a) $5^2 - 2 = 23$. (b) $23^2 - 2 = 529 - 2 = 527$.
2
If $a + b = 10$ and $ab = 21$, find the value of $a^3 + b^3$.
उत्तर एवं व्याख्या देखें
उत्तर: Apply the cubic expansion identity:
$$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$
Substitute the given values $a + b = 10$ and $ab = 21$:
$$10^3 = a^3 + b^3 + 3(21)(10)$$
$$1000 = a^3 + b^3 + 630$$
$$a^3 + b^3 = 1000 - 630 = \mathbf{370}$$.
$a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 10^3 - 3(21)(10) = 1000 - 630 = 370$.
3
Without calculating the cubes directly, evaluate: $(28)^3 + (-15)^3 + (-13)^3$.
उत्तर एवं व्याख्या देखें
उत्तर: Let $a = 28, b = -15, c = -13$.
Step 1: Check the sum $(a + b + c)$:
$$a + b + c = 28 + (-15) + (-13) = 28 - 28 = 0$$
Step 2: Apply the Conditional Identity:
$$\text{If } a + b + c = 0, \quad \text{then } a^3 + b^3 + c^3 = 3abc$$
$$= 3 \times (28) \times (-15) \times (-13)$$
Notice that $(-15) \times (-13) = +195$:
$$= 3 \times 28 \times 195 = 84 \times 195 = \mathbf{16,380}$$.
Since $28 + (-15) + (-13) = 0$, the sum equals $3abc = 3(28)(-15)(-13) = 16,380$.
4
Expand using identity: $(2x - 3y + 4z)^2$.
उत्तर एवं व्याख्या देखें
उत्तर: Apply the Trinomial Square identity: $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.
Here $a = 2x, b = -3y, c = 4z$:
$$= (2x)^2 + (-3y)^2 + (4z)^2 + 2(2x)(-3y) + 2(-3y)(4z) + 2(4z)(2x)$$
$$= \mathbf{4x^2 + 9y^2 + 16z^2 - 12xy - 24yz + 16zx}$$.
Square each term ($4x^2 + 9y^2 + 16z^2$), then add pairwise double products with appropriate signs.
5
Evaluate using algebraic identities without actual multiplication:
(a) $105 \times 95$
(b) $103^2$.
उत्तर एवं व्याख्या देखें
उत्तर:

• (a) $105 \times 95$:
Express as $(100 + 5)(100 - 5)$ and apply $(a + b)(a - b) = a^2 - b^2$:

$$= 100^2 - 5^2 = 10,000 - 25 = \mathbf{9,975}$$


�� (b) $103^2$:
Express as $(100 + 3)^2$ and apply $(a + b)^2 = a^2 + 2ab + b^2$:

$$= 100^2 + 2(100)(3) + 3^2 = 10,000 + 600 + 9 = \mathbf{10,609}$$

.


(a) $(100 + 5)(100 - 5) = 10000 - 25 = 9975$. (b) $(100 + 3)^2 = 10000 + 600 + 9 = 10609$.
6
If $x - \frac{1}{x} = 4$, find the value of $x^3 - \frac{1}{x^3}$.
उत्तर एवं व्याख्या देखें
उत्तर: Apply the cubic reciprocal identity:
$$\left( x - \frac{1}{x} \right)^3 = x^3 - \frac{1}{x^3} - 3\left( x - \frac{1}{x} \right)$$
Substitute $x - \frac{1}{x} = 4$:
$$4^3 = x^3 - \frac{1}{x^3} - 3(4)$$
$$64 = x^3 - \frac{1}{x^3} - 12$$
$$x^3 - \frac{1}{x^3} = 64 + 12 = \mathbf{76}$$.
$x^3 - 1/x^3 = (x - 1/x)^3 + 3(x - 1/x) = 4^3 + 3(4) = 64 + 12 = 76$.
7
Prove the identity: $(a + b)^2 - (a - b)^2 = 4ab$.
उत्तर एवं व्याख्या देखें
उत्तर: Expand both terms on the LHS:
• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$
$$\text{LHS} = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)$$
$$= a^2 + 2ab + b^2 - a^2 + 2ab - b^2$$
$$= (a^2 - a^2) + (b^2 - b^2) + (2ab + 2ab)$$
$$= 0 + 0 + 4ab = \mathbf{4ab = \text{RHS}}$$. Proved!
Expand both squares: $(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab$.
8
If $a^2 + b^2 + c^2 = 29$ and $a + b + c = 9$, find the value of $ab + bc + ca$.
उत्तर एवं व्याख्या देखें
उत्तर: Apply the Trinomial Square identity:
$$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$$
Substitute known values:
$$9^2 = 29 + 2(ab + bc + ca)$$
$$81 = 29 + 2(ab + bc + ca)$$
$$2(ab + bc + ca) = 81 - 29 = 52$$
$$ab + bc + ca = \frac{52}{2} = \mathbf{26}$$.
$2(ab + bc + ca) = (a + b + c)^2 - (a^2 + b^2 + c^2) = 81 - 29 = 52 \implies ab + bc + ca = 26$.
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