MP Board Class 8 Mathematics

Quick Revision Notes & Formula Sheet

Chapter: Algebraic Expressions and Identities (बीजीय व्यंजक और सर्वसमिकाएँ)

---

1. Basic Concepts (मूल अवधारणाएं)

• Variable (चर): A symbol that does not have a fixed value. It can take various values (denoted by letters like $x, y, z, a, b, c$).
• Constant (अचर): A quantity having a fixed numerical value (e.g., $3, -5, \frac{1}{2}, 100$).
• Algebraic Expression (बीजीय व्यंजक): A combination of variables and constants connected by fundamental mathematical operators ($+$, $-$, $\times$, $\div$).
  • Example: $3x^2 + 5y - 7$

---

2. Components of an Expression

For an expression like $4x^2 - 3xy + 8$:

1. Terms (पद): Parts of an expression formed separately and then added together.
   • Terms in $4x^2 - 3xy + 8$ are: $4x^2$, $-3xy$, and $8$.
2. Factors (गुणनखंड): Quantities multiplied together to form a term.
   • Factors of $-3xy$ are: $-3$, $x$, and $y$.
3. Coefficient (गुणांक): The numerical factor of a term.
   • Coefficient of $x^2$ in $4x^2$ is: $4$.
   • Coefficient of $xy$ in $-3xy$ is: $-3$.

---

3. Types of Expressions (व्यंजकों के प्रकार)

Type	Definition	Example
Monomial (एकपदी)	An expression containing 1 term.	$5x$, $-7$, $3ab$
Binomial (द्विपदी)	An expression containing 2 terms.	$a + b$, $4x - 3y$
Trinomial (त्रिपदी)	An expression containing 3 terms.	$x + y + z$, $2a^2 - 3a + 5$
Polynomial (बहुपद)	An expression containing one or more terms with non-zero coefficients and non-negative integral exponents.	$a + b + c + d$, $x^3 - 4x^2 + 2x - 7$

---

4. Like and Unlike Terms (सजातीय और विजातीय पद)

• Like Terms (सजातीय पद): Terms having the same algebraic variables with the same exponents.
  • Examples: $7x$ and $-12x$ | $3x^2y$ and $-5x^2y$
  • Note: Only like terms can be added or subtracted!
• Unlike Terms (विजातीय पद): Terms having different variables or different exponents.
  • Examples: $7x$ and $7y$ | $3x^2$ and $3x^3$

---

5. Key Rules for Operations

A. Addition & Subtraction (योग एवं घटाव)

1. Group the like terms together.
2. Add or subtract their numerical coefficients.
3. Subtraction Rule: Change the sign of each term of the expression being subtracted.
   • -(a + b - c) = -a - b + c

B. Multiplication (गुणा के नियम)

1. Sign Rules:
   • $(+) \times (+) = (+)$
   • $(-) \times (-) = (+)$
   • $(+) \times (-) = (-)$
   • $(-) \times (+) = (-)$
2. Exponent Law for Multiplication:
   • $x^m \times x^n = x^{(m + n)}$
3. Distributive Law:
   • $a \times (b + c) = ab + ac$
   • $(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$

---

6. Standard Algebraic Identities (मानक सर्वसमिकाएँ)

An Identity is an equality that is true for all values of the variables in it.

Identity I

$$(a + b)^2 = a^2 + 2ab + b^2$$

Identity II

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

Identity III

$$(a + b)(a - b) = a^2 - b^2$$

Identity IV

$$(x + a)(x + b) = x^2 + (a + b)x + ab$$

---

7. Application of Identities (Shortcuts for Calculations)

Identities can be used to easily calculate squares and products of numbers:

1. Using Identity I:

   • $103^2 = (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609$

2. Using Identity II:

   • $99^2 = (100 - 1)^2 = 100^2 - 2(100)(1) + 1^2 = 10000 - 200 + 1 = 9801$

3. Using Identity III:

   • $103 \times 97 = (100 + 3)(100 - 3) = 100^2 - 3^2 = 10000 - 9 = 9991$

4. Using Identity IV:

   • $501 \times 502 = (500 + 1)(500 + 2) = 500^2 + (1 + 2)(500) + (1 \times 2)$
   • $= 250000 + 1500 + 2 = 251502$

---

8. Quick Exam Tips for MP Board

• Always write the Identity formula first before solving identity-based questions to get full steps marks.
• Be careful with negative signs, especially during subtraction and multiplication.
• Remember: An Equation is true only for specific values of variables, whereas an Identity is true for all values.