Class 8 — Mathematics (NCERT)

Algebraic Expressions and Identities: Multiplication and Standard Identities

Class 8

✓By the end of this lesson students will be able to multiply a monomial by a monomial, a monomial by a polynomial, and a polynomial by a polynomial.
✓By the end of this lesson students will be able to understand and apply the three standard algebraic identities.
✓By the end of this lesson students will be able to simplify algebraic expressions using appropriate multiplication methods and identities.
✓By the end of this lesson students will be able to solve problems involving multiplication of algebraic expressions and standard identities.

Key concepts

Multiplication of Monomial by Monomial

To multiply two monomials, we multiply their numerical coefficients and then multiply their variable parts. Remember the law of exponents: x^m × x^n = x^(m+n). For example, (3x) × (5y) = (3 × 5) × (x × y) = 15xy. Also, (2x²) × (4x³) = (2 × 4) × (x² × x³) = 8x^(2+3) = 8x⁵.

Multiplication of Monomial by Polynomial

When a monomial is multiplied by a polynomial, each term of the polynomial is multiplied by the monomial. This is based on the distributive property: a(b + c) = ab + ac. For example, 2x(3x + 5y) = (2x × 3x) + (2x × 5y) = 6x² + 10xy.

a(b + c) = ab + ac

Multiplication of Polynomial by Polynomial

To multiply two polynomials, we multiply each term of the first polynomial by each term of the second polynomial. Then, we combine the like terms. For example, (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd.

(a + b)(c + d) = ac + ad + bc + bd

Standard Identity I: (a + b)²

This identity states that the square of the sum of two terms is equal to the square of the first term, plus twice the product of the two terms, plus the square of the second term. We can derive it by multiplying (a + b) by (a + b): (a + b)² = (a + b)(a + b) = a(a + b) + b(a + b) = a² + ab + ba + b² = a² + 2ab + b².

(a + b)² = a² + 2ab + b²

Standard Identity II: (a - b)²

This identity states that the square of the difference of two terms is equal to the square of the first term, minus twice the product of the two terms, plus the square of the second term. We can derive it by multiplying (a - b) by (a - b): (a - b)² = (a - b)(a - b) = a(a - b) - b(a - b) = a² - ab - ba + b² = a² - 2ab + b².

(a - b)² = a² - 2ab + b²

Standard Identity III: (a + b)(a - b)

This identity states that the product of the sum and difference of two terms is equal to the difference of their squares. We can derive it by multiplying (a + b) by (a - b): (a + b)(a - b) = a(a - b) + b(a - b) = a² - ab + ba - b² = a² - b².

(a + b)(a - b) = a² - b²

Key facts to remember

1To multiply monomials, multiply coefficients and add powers of like variables.
2The distributive property is fundamental for multiplying a monomial by a polynomial: a(b + c) = ab + ac.
3To multiply two polynomials, multiply each term of the first polynomial by each term of the second polynomial, then combine like terms.
4Standard Identity I: (a + b)² = a² + 2ab + b²
5Standard Identity II: (a - b)² = a² - 2ab + b²
6Standard Identity III: (a + b)(a - b) = a² - b²
7Algebraic identities are equalities that are true for all values of the variables involved.
8Identities provide shortcuts for expanding and simplifying algebraic expressions and for performing numerical calculations.

Worked examples

Example 1

Multiply (2x + 3y) by (4x - 5y).

IWe need to multiply each term of the first polynomial by each term of the second polynomial.

II(2x + 3y)(4x - 5y) = 2x(4x - 5y) + 3y(4x - 5y)

III= (2x × 4x) - (2x × 5y) + (3y × 4x) - (3y × 5y)

IV= 8x² - 10xy + 12xy - 15y²

VCombine the like terms (-10xy and +12xy):

VI= 8x² + 2xy - 15y²

Answer

8x² + 2xy - 15y²

Remember to multiply coefficients and add powers of variables for each term.

Example 2

Expand (3a + 4b)² using a suitable identity.

IWe use the Standard Identity I: (x + y)² = x² + 2xy + y².

IIIn our problem, x = 3a and y = 4b.

IIISubstitute these values into the identity:

IV(3a + 4b)² = (3a)² + 2(3a)(4b) + (4b)²

V= (3² × a²) + (2 × 3 × 4 × a × b) + (4² × b²)

VI= 9a² + 24ab + 16b²

Answer

9a² + 24ab + 16b²

Ensure you square both the numerical coefficient and the variable part.

Example 3

Evaluate 52 × 48 using a suitable identity.

IWe can express 52 as (50 + 2) and 48 as (50 - 2).

IISo, 52 × 48 = (50 + 2)(50 - 2).

IIIThis expression is in the form of Standard Identity III: (a + b)(a - b) = a² - b².

IVHere, a = 50 and b = 2.

VSubstitute these values into the identity:

VI(50 + 2)(50 - 2) = (50)² - (2)²

VII= 2500 - 4

VIII= 2496

Answer

2496

Using identities can simplify numerical calculations significantly.

Example 4

Simplify (2x - 3y)² - (2x + 3y)².

IWe will use Standard Identity II: (a - b)² = a² - 2ab + b² for the first term.

IIAnd Standard Identity I: (a + b)² = a² + 2ab + b² for the second term.

IIIFor (2x - 3y)²: Let a = 2x, b = 3y.

IV(2x - 3y)² = (2x)² - 2(2x)(3y) + (3y)² = 4x² - 12xy + 9y².

VFor (2x + 3y)²: Let a = 2x, b = 3y.

VI(2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y².

VIINow, substitute these expanded forms back into the original expression:

VIII(4x² - 12xy + 9y²) - (4x² + 12xy + 9y²)

9Carefully distribute the negative sign to all terms inside the second parenthesis:

10= 4x² - 12xy + 9y² - 4x² - 12xy - 9y²

11Combine the like terms:

12= (4x² - 4x²) + (-12xy - 12xy) + (9y² - 9y²)

13= 0 - 24xy + 0

14= -24xy

Answer

-24xy

Be extremely careful with signs, especially when subtracting an entire expanded expression.

Common mistakes

✗Incorrectly applying the distributive property, especially when multiplying polynomials.
✗Making sign errors during multiplication, particularly when dealing with negative terms.
✗Forgetting to add the powers of variables when multiplying terms with the same base (e.g., x² × x³ ≠ x⁶).
✗Incorrectly expanding (a + b)² as a² + b² or (a - b)² as a² - b².
✗Not identifying 'a' and 'b' correctly in the standard identities, especially when terms are complex or involve negative signs.

Exam tips

★Memorise the three standard algebraic identities thoroughly. This will save time and prevent errors.
★Practice the distributive property extensively with various types of expressions to build accuracy.
★Always pay close attention to the signs of the terms when multiplying and combining like terms.
★Show all steps clearly in your exam answers. This helps in identifying errors and earns partial marks even if the final answer is incorrect.
★When simplifying expressions involving multiple identities, expand each part separately and then combine them, being careful with subtraction.

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