An algebraic expression is built by combining variables (unknown letters like $x, y$) and constants (numbers like $4, -7$) through addition, subtraction, multiplication, and division. An algebraic identity is an equality that remains true for all conceivable values of its variables, unlike an algebraic equation which holds true only for specific root values. NCERT Class 8 Mathematics Chapter 8 covers term factorization, coefficient identification, polynomial arithmetic, and the mastery of the four standard identities for rapid algebraic and numerical computation.
High-Yield Chapter Key Takeaways
- Term Anatomy: In a term like $-5x^2y$, $-5$ is the numerical coefficient, while $x, x, y$ are the algebraic (literal) factors.
- Only Like Terms Combine: Addition and subtraction are strictly permitted between like terms (same variables with identical powers). Unlike terms remain separated.
- Polynomial Exponent Constraint: Expressions containing negative or fractional powers (such as $\frac{1}{x} = x^{-1}$ or $\sqrt{x} = x^{1/2}$) are not polynomials.
- Identity vs. Equation: $(a + 1)(a + 2) = a^2 + 3a + 2$ is an identity (valid for all $a$), whereas $x^2 + 3x + 2 = 12$ is an equation (valid only for specific roots $x = 2, -5$).
- Mental Math Power: Identities allow rapid evaluation of large numbers, e.g., $103 \times 97 = (100 + 3)(100 - 3) = 10000 - 9 = 9991$.
1. What is an Algebraic Expression?
In arithmetic, we deal with known numerical values like $5, 12, -9$. In algebra, we encounter situations where certain quantities are not known beforehand or can change. We denote these unknown or varying quantities using variables (commonly letters of the alphabet such as $x, y, z, a, b, l, m$).
When constants and variables are combined using arithmetic operations ($+, -, \times, \div$), we obtain an algebraic expression.
- $2x + 3$: Variable $x$ multiplied by 2, then increased by 3.
- $4y - 7$: Variable $y$ multiplied by 4, then decreased by 7.
- $x^2 + 2xy + y^2$: A multi-variable expression involving squares and products.
- $x + 5$ is located 5 units to the right of $X$.
- $x - 4$ is located 4 units to the left of $X$.
- $2x + 1$ is obtained by first doubling the distance $OX$ and then shifting 1 unit to the right.
2. Terms, Factors, and Numerical Coefficients
To work with expressions methodically, we break them down into their constituent building blocks: terms, factors, and coefficients.
- Terms: The individual parts of an expression that are separated by $+$ or $-$ signs. In the expression $4x^2 - 3xy + 8$, the terms are $4x^2$, $-3xy$, and $8$. (Always carry the preceding negative sign with the term!).
- Factors: Each term is a product of factors. For example, the term $4x^2$ is the product of $4$, $x$, and $x$. The term $-3xy$ is the product of $-3$, $x$, and $y$.
- Numerical Coefficient: The numerical factor multiplying the literal variables in a term. In $-3xy$, the numerical coefficient is $-3$. When no number appears before a variable (like in $y$ or $x^2$), the coefficient is understood to be $1$. In $-x$, the coefficient is $-1$.
| Expression | Term | Factors | Numerical Coefficient |
|---|---|---|---|
| $7xy - 5x$ | $7xy$ | $7, x, y$ | $+7$ |
| $-5x$ | $-5, x$ | $-5$ | |
| $1 - 2y + 3y^2$ | $1$ | $1$ | $1$ (Constant term) |
| $-2y$ | $-2, y$ | $-2$ | |
| $3y^2$ | $3, y, y$ | $+3$ | |
| $\frac{x}{2} - \frac{y}{3}$ | $\frac{1}{2}x$ | $\frac{1}{2}, x$ | $+\frac{1}{2}$ |
| $-\frac{1}{3}y$ | $-\frac{1}{3}, y$ | $-\frac{1}{3}$ | |
| $0.3a - 0.6ab$ | $0.3a$ | $0.3, a$ | $+0.3$ |
| $-0.6ab$ | $-0.6, a, b$ | $-0.6$ |
3. Classification of Expressions: Monomials, Binomials, Trinomials & Polynomials
Algebraic expressions are classified according to the count of unlike terms they contain:
-
Monomial: An expression containing exactly one term.
Examples: $4x^2$, $7xyz$, $-9$, $3l^2m$. -
Binomial: An expression containing exactly two unlike terms.
Examples: $a + b$, $4l + 5m$, $2x - 3y$, $p^2 - q^2$. -
Trinomial: An expression containing exactly three unlike
terms.
Examples: $a + b + c$, $2x + 3y - 5$, $x^2 - 5x + 6$. - Polynomial: An algebraic expression containing one or more terms with non-negative integer exponents on all variables and non-zero coefficients.
For an expression to be a polynomial, all exponents of the variables MUST be whole numbers ($0, 1, 2, 3, \dots$).
- $\frac{1}{x} + 3 = x^{-1} + 3$ is NOT a polynomial because the exponent $-1$ is negative.
- $\sqrt{x} + 5 = x^{1/2} + 5$ is NOT a polynomial because the exponent $\frac{1}{2}$ is a fraction.
- $\frac{2}{y^2} - y$ is NOT a polynomial because $y^{-2}$ has a negative exponent.
4. Like and Unlike Terms
Understanding the distinction between like and unlike terms is fundamental to all algebraic manipulations:
- Like Terms: Terms that have the identical algebraic (literal)
factors. Their numerical coefficients may be completely different, and the order of
multiplication of variables does not matter ($xy = yx$).
Examples: $7x$ and $14x$; $-5x^2y$ and $3yx^2$; $\frac{1}{2}ab$ and $-7ba$. - Unlike Terms: Terms that differ either in the variables involved or in the powers
(exponents) to which those variables are raised.
Examples: $7x$ and $7y$ (different variables); $4x^2y$ and $4xy^2$ (in the first, $x$ is squared; in the second, $y$ is squared).
5. Addition and Subtraction of Algebraic Expressions
There are two primary methods for adding and subtracting expressions: the Horizontal Method and the Column Method.
Addition of Expressions
To add expressions, collect like terms together and add their numerical coefficients.
Sum $= (7x^2 - 4x + 5) + (9x - 10) + (4x^2 - 3x + 7)$
Group like terms: $= (7x^2 + 4x^2) + (-4x + 9x - 3x) + (5 - 10 + 7)$
Combine coefficients: $= (7 + 4)x^2 + (-4 + 9 - 3)x + (12 - 10)$
$= 11x^2 + 2x + 2$
Method 2 (Column Method):
Align like terms vertically:
$\phantom{+ 0}7x^2 - 4x + 5$
$\phantom{+ 0000} + 9x - 10$
$\underline{+ 4x^2 - 3x + 7}$
$= 11x^2 + 2x + 2$
Subtraction of Expressions (The Sign-Flip Rule)
When subtracting expression $B$ from expression $A$, remember that subtracting a number is the same as adding its additive inverse. Therefore, every term in the expression being subtracted has its sign reversed (+ becomes -, and - becomes +).
Write the minuend on top and subtrahend underneath with aligned like terms:
$\phantom{- (}12a - 9ab + 5b - 3$
$\phantom{- (}\phantom{1}4a - 7ab + 3b + 12$
$\underline{\phantom{-(}(-) \quad (+) \quad (-) \quad (-)\phantom{0000}}$ (Reversed signs of lower row)
$= (12 - 4)a + (-9 + 7)ab + (5 - 3)b + (-3 - 12)$
$= 8a - 2ab + 2b - 15$
6. Multiplication of Algebraic Expressions
Multiplication in algebra relies on the commutative and associative laws of real numbers together with the laws of exponents ($x^m \cdot x^n = x^{m+n}$).
A. Multiplying a Monomial by a Monomial
- Multiply the numerical coefficients first.
- Multiply matching algebraic variables by adding their exponents ($x^a \cdot x^b = x^{a+b}$).
- Group distinct variables side-by-side alphabetically.
Examples:
- $(3x) \times (5y) = (3 \times 5)(x \times y) = 15xy$
- $(-4x^2) \times (3x^3) = [(-4) \times 3](x^{2+3}) = -12x^5$
- $(2x) \times (3y) \times (5z) = (2 \times 3 \times 5)(x \times y \times z) = 30xyz$
B. Multiplying a Monomial by a Polynomial
Apply the Distributive Law of Multiplication over Addition: $a(b + c) = ab + ac$.
- $3x(2x + 5y) = (3x \times 2x) + (3x \times 5y) = 6x^2 + 15xy$
- $7a^2(2ab - 5c) = (7a^2 \times 2ab) - (7a^2 \times 5c) = 14a^3b - 35a^2c$
C. Multiplying a Binomial by a Binomial (FOIL Method)
Multiply each term of the first binomial by every term of the second binomial: $$(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$$
$= (2x \cdot 3x) + (2x \cdot 4y) + (3y \cdot 3x) + (3y \cdot 4y)$
$= 6x^2 + 8xy + 9xy + 12y^2$
Combine like terms ($8xy + 9xy = 17xy$):
$= 6x^2 + 17xy + 12y^2$
7. What is an Identity? (Identity vs. Equation)
Students often confuse an algebraic identity with an algebraic equation. The distinction is critical:
| Criterion | Algebraic Identity | Algebraic Equation |
|---|---|---|
| Definition | An equality that is valid for all values of the variable(s). | An equality that is valid only for specific value(s) of the variable (its roots). |
| Sample Equality | $(a + 1)(a + 2) = a^2 + 3a + 2$ | $x^2 + 3x + 2 = 12$ |
| Test $x = 1$ | LHS $= 2 \times 3 = 6$; RHS $= 1 + 3 + 2 = 6$. (LHS = RHS ✓) | LHS $= 1 + 3 + 2 = 6 \neq 12$. (LHS $\neq$ RHS ✗) |
| Test $x = 5$ | LHS $= 6 \times 7 = 42$; RHS $= 25 + 15 + 2 = 42$. (LHS = RHS ✓) | LHS $= 25 + 15 + 2 = 42 \neq 12$. (LHS $\neq$ RHS ✗) |
| Solution Count | Infinitely many solutions (holds universally). | Finite roots (here, only $x = 2$ and $x = -5$). |
8. The Four Standard Algebraic Identities
In NCERT Class 8, four standard algebraic identities are introduced. These identities must be thoroughly memorized and recognized in both forward (expansion) and reverse (factorization) directions.
The square of the sum of two terms equals the square of the first, plus twice their product, plus the square of the second.
The square of a difference equals the square of the first, minus twice their product, plus the square of the second. Notice that the last term is $+b^2$.
The product of the sum and difference of two terms equals the difference between their individual squares.
Multiplying two binomials sharing a common first term $x$ yields $x^2$, plus the sum of constants times $x$, plus the product of constants.
Proofs and Derivations:
- Identity I: $(a + b)^2 = (a + b)(a + b) = a(a + b) + b(a + b) = a^2 + ab + ba + b^2
= a^2 + 2ab + b^2$.
Geometric Meaning: A square of side $(a + b)$ has area $(a+b)^2$, which subdivides into a square of area $a^2$, a square of area $b^2$, and two identical rectangles of area $ab$. - Identity II: $(a - b)^2 = (a - b)(a - b) = a(a - b) - b(a - b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2$.
- Identity III: $(a + b)(a - b) = a(a - b) + b(a - b) = a^2 - ab + ba - b^2 = a^2 - b^2$.
- Identity IV: $(x + a)(x + b) = x(x + b) + a(x + b) = x^2 + bx + ax + ab = x^2 + (a + b)x + ab$.
- $(x - a)(x + b) = x^2 + (b - a)x - ab$
- $(x + a)(x - b) = x^2 + (a - b)x - ab$
- $(x - a)(x - b) = x^2 - (a + b)x + ab$
9. Fast Mental Math Using Identities (CBSE Favorite Section)
Examiners frequently test students on computing arithmetic products without using long multiplication by decomposing numbers into base-10 sums or differences.
Write $103 = (100 + 3)$. Apply Identity I with $a = 100, b = 3$:
$103^2 = (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2$
$= 10000 + 600 + 9 = \mathbf{10609}$
(ii) Evaluate $98^2$:
Write $98 = (100 - 2)$. Apply Identity II with $a = 100, b = 2$:
$98^2 = (100 - 2)^2 = 100^2 - 2(100)(2) + 2^2$
$= 10000 - 400 + 4 = 9600 + 4 = \mathbf{9604}$
(iii) Evaluate $105 \times 95$:
Write $105 = (100 + 5)$ and $95 = (100 - 5)$. Apply Identity III:
$105 \times 95 = (100 + 5)(100 - 5) = 100^2 - 5^2$
$= 10000 - 25 = \mathbf{9975}$
(iv) Evaluate $103 \times 104$:
Write $103 = (100 + 3)$ and $104 = (100 + 4)$. Apply Identity IV with $x = 100, a = 3, b = 4$:
$103 \times 104 = (100 + 3)(100 + 4) = 100^2 + (3 + 4)(100) + (3 \times 4)$
$= 10000 + 700 + 12 = \mathbf{10712}$
10. Common CBSE Exam Traps & Mistakes
Top Mistakes Students Make in Exam Papers
- Omitting the Middle Term $2ab$: Writing $(a + b)^2 = a^2 + b^2$ is one of the most widespread errors. Always remember the cross-product term $+2ab$.
- Incorrect Sign in Identity II: Writing $(a - b)^2 = a^2 - 2ab - b^2$. Remember that $(-b)^2 = +b^2$, so the last term is strictly positive ($+b^2$).
- Combining Unlike Terms: Treating $2x + 3y$ as $5xy$ or $3x^2 + 2x$ as $5x^3$. You can only add coefficients of terms with identical variable factors.
- Squaring a Monomial Incorrectly: Writing $(3x)^2 = 3x^2$. The coefficient must also be squared: $(3x)^2 = 3^2 \cdot x^2 = 9x^2$.
- Sign Reversal in Subtraction: When asked to "subtract $P$ from $Q$", students write $P - Q$ instead of $Q - P$. Moreover, failing to distribute the negative sign across all terms inside the parentheses leads to full marks loss.
11. Solved High-Yield Exemplar Problems
First term: $(4m + 5n)^2 = (4m)^2 + 2(4m)(5n) + (5n)^2 = 16m^2 + 40mn + 25n^2$
Second term: $(5m + 4n)^2 = (5m)^2 + 2(5m)(4n) + (4n)^2 = 25m^2 + 40mn + 16n^2$
Now add the two expansions together:
$= (16m^2 + 25m^2) + (40mn + 40mn) + (25n^2 + 16n^2)$
$= 41m^2 + 80mn + 41n^2$
Squaring both sides of the equation:
$$\left(x + \frac{1}{x}\right)^2 = 5^2$$ Using Identity I $(a + b)^2 = a^2 + 2ab + b^2$ with $a = x$ and $b = \frac{1}{x}$:
$$x^2 + 2(x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = 25$$ $$x^2 + 2(1) + \frac{1}{x^2} = 25$$ $$x^2 + \frac{1}{x^2} = 25 - 2$$ $$x^2 + \frac{1}{x^2} = \mathbf{23}$$
$a^2 - b^2 = (a + b)(a - b)$
$= (1.02 + 0.98) \times (1.02 - 0.98)$
$= (2.00) \times (0.04)$
$= \mathbf{0.08}$
12. Frequently Asked Questions (Class 8 Algebra)
What is an algebraic expression and what are its components?
- Terms: Elements separated by $+$ or $-$ signs (e.g., in $4x^2 - 3xy$, the terms are $4x^2$ and $-3xy$).
- Factors: Quantities multiplied together to form a term (e.g., $4, x, x$ are factors of $4x^2$).
- Numerical Coefficient: The constant number multiplying the variable in a term (e.g., $-3$ is the coefficient of $-3xy$).
What is the difference between like and unlike terms?
What is the difference between an identity and an equation?
Why is $\frac{1}{x} + 5$ not considered a polynomial?
What are the four standard algebraic identities in NCERT Class 8?
- $(a + b)^2 = a^2 + 2ab + b^2$
- $(a - b)^2 = a^2 - 2ab + b^2$
- $(a + b)(a - b) = a^2 - b^2$
- $(x + a)(x + b) = x^2 + (a + b)x + ab$
How can standard identities be used for quick arithmetic evaluation?
- $103^2 = (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609$
- $105 \times 95 = (100 + 5)(100 - 5) = 100^2 - 5^2 = 10000 - 25 = 9975$
- $99 \times 101 = (100 - 1)(100 + 1) = 100^2 - 1 = 9999$
