NCERT Solutions for Class 8 Maths Chapter 8: Algebraic Expressions and Identities — Free PDF

Chapter 8 is one of the most algebra-heavy chapters in Class 8 and forms the backbone of algebraic manipulation for Classes 9 and 10. You will learn to classify expressions as monomials, binomials, trinomials, and polynomials, perform addition, subtraction, and multiplication of algebraic expressions, and master the four standard algebraic identities.

These identities are not just formulas to memorise — they are powerful tools that allow you to expand expressions instantly, simplify complex calculations mentally, and later factorise polynomials. Getting comfortable with them now is essential for your future success in algebra, coordinate geometry, and even calculus.

The chapter has 4 exercises with a total of around 25 problems. Exercises 8.1 and 8.2 focus on addition, subtraction, and multiplication of algebraic expressions. Exercises 8.3 and 8.4 are dedicated to applying algebraic identities for expansion and numerical computation. Every problem in Chapters 12 (Factorisation) and 14 (Equations) uses skills from this chapter.

1. Algebraic Expression: A combination of constants, variables, and arithmetic operations ($+, -, \times, \div$). For example, $3x^2 + 5x - 7$ is an algebraic expression.

2. Term: Each part of an expression separated by $+$ or $-$ signs. In $4x^2 - 3xy + 7$, the terms are $4x^2$, $-3xy$, and $7$.

3. Coefficient: The numerical factor of a term. In $-5x^2y$, the coefficient is $-5$.

4. Types of Expressions:
- Monomial: One term (e.g., $3x^2$, $-7ab$, $5$)
- Binomial: Two terms (e.g., $2x + 5$, $a^2 - b^2$)
- Trinomial: Three terms (e.g., $x^2 + 3x + 2$)
- Polynomial: One or more terms with non-negative integer powers of variables.

5. Like and Unlike Terms: Like terms have the same variable parts (e.g., $3x^2$ and $-5x^2$). Unlike terms have different variable parts (e.g., $3x^2$ and $3x$). Only like terms can be added or subtracted.

6. Degree of a Polynomial: The highest power of the variable in the polynomial. For example, $4x^3 + 2x - 1$ has degree $3$.

7. Important: $x^2$ and $x$ are NOT like terms. $3xy$ and $3yx$ ARE like terms (multiplication is commutative). The coefficient includes the sign ($-7x$ has coefficient $-7$).

Multiplication of algebraic expressions follows three patterns:

**1. Monomial $\times$ Monomial:**
Multiply the coefficients and add the exponents of like variables.

**2. Monomial $\times$ Polynomial:**
Distribute the monomial to each term of the polynomial.

**3. Polynomial $\times$ Polynomial:**
Multiply each term of one polynomial by each term of the other, then combine like terms.

These four identities must be memorised perfectly. They are used throughout Classes 8, 9, and 10.

Identity I:

Squaring a sum gives three terms: the square of the first, twice the product, and the square of the second.

Identity II:

Squaring a difference also gives three terms, but the middle term is negative.

Identity III:

The product of the sum and difference of two terms equals the difference of their squares.

Identity IV:

This is the general product of two binomials with the same first term.

Critical Warning: $(a + b)^2 \neq a^2 + b^2$. The middle term $2ab$ must never be forgotten. This is the single most common error in all of algebra.

**Q1. Add: $3x^2 + 5x - 4$ and $2x^2 - 3x + 7$.**

Solution:

Combine like terms:
- $x^2$ terms: $3x^2 + 2x^2 = 5x^2$
- $x$ terms: $5x - 3x = 2x$
- Constants: $-4 + 7 = 3$

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**Q2. Subtract $4a^2 - 7a + 3$ from $6a^2 + 2a - 5$.**

Solution:

Change the sign of every term in the second expression:

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**Q3. Multiply: $3x^2y \times (-2xy^3)$.**

Solution:

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**Q4. Multiply: $2x(3x^2 - 5x + 4)$.**

Solution:

Distribute $2x$ to each term:

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**Q5. Multiply: $(3x + 2)(2x - 5)$.**

Solution:

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**Q6. Find the product: $(a + 2b)(a^2 - 2ab + 4b^2)$.**

Solution:

This is actually the identity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ with $b = 2b$.

**Q1. Find $(3x + 4y)^2$ using Identity I.**

Solution:

Using $(a+b)^2 = a^2 + 2ab + b^2$ with $a = 3x, b = 4y$:

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**Q2. Evaluate $99^2$ using Identity II.**

Solution:

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**Q3. Find $103 \times 97$ using Identity III.**

Solution:

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**Q4. Find $(x + 3)(x + 7)$ using Identity IV.**

Solution:

Using $(x+a)(x+b) = x^2 + (a+b)x + ab$ with $a = 3, b = 7$:

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**Q5. Expand $(2a - 3b)^2$ using Identity II.**

Solution:

Using $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 2a, b = 3b$:

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**Q6. Evaluate $104^2$ using Identity I.**

Solution:

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**Q7. Find $(x + 4)(x - 3)$ using Identity IV.**

Solution:

Using $(x+a)(x+b) = x^2 + (a+b)x + ab$ with $a = 4, b = -3$:

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**Q8. Simplify: $(5x + 3y)^2 - (5x - 3y)^2$.**

Solution:

Using Identities I and II:

Subtracting:

Alternatively, use Identity III directly: $(A + B)(A - B) = A^2 - B^2$ where $A = 5x+3y$ and $B = 5x-3y$... but actually, a quicker approach is to note $(a+b)^2 - (a-b)^2 = 4ab$. With $a = 5x, b = 3y$: answer $= 4(5x)(3y) = 60xy$.

Here are more examples covering common exam patterns.

**Example 1. Evaluate $48 \times 52$ using an identity.**

Solution:

This uses Identity III: $(a-b)(a+b) = a^2 - b^2$.

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**Example 2. If $x + \dfrac{1}{x} = 5$, find $x^2 + \dfrac{1}{x^2}$.**

Solution:

Square both sides using Identity I:

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**Example 3. If $a - b = 3$ and $a^2 + b^2 = 29$, find $ab$.**

Solution:

Using Identity II: $(a - b)^2 = a^2 - 2ab + b^2$

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**Example 4. Using identities, evaluate $10.2 \times 9.8$.**

Solution:

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**Example 5. Without multiplying directly, find $(3x + 5)^2 + (3x - 5)^2$.**

Solution:

Using the identity $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$:

Students frequently lose marks in this chapter due to these errors:

**Mistake 1: Writing $(a+b)^2 = a^2 + b^2$.**
This is the most common algebra mistake across all classes. Always remember: $(a+b)^2 = a^2 + 2ab + b^2$. Squaring a binomial produces THREE terms, not two.

Mistake 2: Sign errors during subtraction.
When subtracting one expression from another, you must change the sign of EVERY term in the expression being subtracted. Missing even one sign change leads to a wrong answer.

Mistake 3: Mixing up like and unlike terms.
$3x^2$ and $3x$ are NOT like terms (different powers). $5xy$ and $-2xy$ ARE like terms. Only like terms can be added or subtracted.

Mistake 4: Forgetting to combine like terms after multiplication.
After multiplying two polynomials, always scan for like terms that need to be combined. Students often leave the answer with uncollected terms.

Mistake 5: Confusing Identity III and Identity IV.
Identity III: $(a+b)(a-b) = a^2 - b^2$ requires the same terms with opposite signs. Identity IV: $(x+a)(x+b) = x^2 + (a+b)x + ab$ is for two binomials with the same first term but different second terms.

Test your understanding with these additional questions.

Q1. Expand $(4x - 5y)^2$.

Answer: $(4x)^2 - 2(4x)(5y) + (5y)^2 = 16x^2 - 40xy + 25y^2$.

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Q2. Evaluate $997^2$ using an identity.

Answer: $997^2 = (1000 - 3)^2 = 1000000 - 6000 + 9 = 994009$.

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Q3. If $a + b = 7$ and $ab = 12$, find $a^2 + b^2$.

Answer: $(a+b)^2 = a^2 + 2ab + b^2$. So $49 = a^2 + b^2 + 24$. Therefore $a^2 + b^2 = 25$.

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Q4. Multiply $(2x + 3y)(2x + 3y)(2x - 3y)$.

Answer: First multiply the last two: $(2x+3y)(2x-3y) = 4x^2 - 9y^2$. Then: $(2x+3y)(4x^2 - 9y^2) = 8x^3 - 18xy^2 + 12x^2y - 27y^3$. But actually, let us be careful: $(2x+3y)^2(2x-3y) = (4x^2 + 12xy + 9y^2)(2x - 3y)$. Expanding: $8x^3 - 12x^2y + 24x^2y - 36xy^2 + 18xy^2 - 27y^3 = 8x^3 + 12x^2y - 18xy^2 - 27y^3$.

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Q5. Find $(x + 2)(x + 5)$ and $(x + 2)(x - 5)$ using Identity IV.

Answer: $(x+2)(x+5) = x^2 + 7x + 10$. $(x+2)(x-5) = x^2 + (2-5)x + (2)(-5) = x^2 - 3x - 10$.

• Algebraic expressions are classified by the number of terms: monomial ($1$), binomial ($2$), trinomial ($3$).
  - Add/subtract like terms only; multiply by distributing each term.
  - The four standard identities are essential tools for expansion, simplification, and mental computation.
  - $(a+b)^2 \neq a^2 + b^2$ — never forget the middle term $2ab$.
  - For numerical calculations, express numbers as sums or differences of round numbers and apply identities.
  - Derived identities: $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$ and $(a+b)^2 - (a-b)^2 = 4ab$.
  - These identities are the foundation for Chapter 12 (Factorisation), which uses them in reverse.
  - When multiplying polynomials, use the table/grid method to ensure no term is missed.

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