Multiplication of Algebraic Expressions

While multiplying two algebraic expressions, each term in one expression is multiplied with each term in other expression.

When two or more terms are multiplied with each other
↬ Coefficients of each term are multiplied as usual, and
↬ Powers of same variable are added.
(∵ a^maⁿ = a^m+n)

3x²y  ⋅ 2xy³ 
= 3⋅x²⋅y  ⋅ 2⋅x⋅y³ 
= 3⋅2⋅x²⋅x⋅y⋅y³ 
= 6x²⁺¹y¹⁺³
= 6x³y⁴

(*An algebraic term itself is a product of constants and variables)

Multiplying two or more monomials

 xy× yz× zx 
= x⋅x⋅y⋅y⋅z⋅z
= x⁽¹⁺¹⁾y⁽¹⁺¹⁾z⁽¹⁺¹⁾
= x²y²z²

a×(– a)²× a³ 
= a× a²× a³ 
= a^(1+2+3)
= a⁶

2×4y×8y³×16y³
= 2×4×8×16×y^(1+3+3)
= 1024y⁷

Product of monomials is also a monomial.

Multiplying a monomial by a polynomial 

We use distributive law of multiplication over addition in multiplying polynomials.
a(b±c) = ab ± ac

2x (3x + 5xy)
= 2x⋅3x + 2x⋅5xy
= 6x² + 10x²y

a² (2ab – 5c)
= a²⋅2ab − a²⋅5c
= 2a³b − 5a²c

(4p²+ 5p + 7) × 3p 
= 4p² × 3p +  5p × 3p + 7× 3p
= 12p³+  15p²+  21p

Multiplying polynomials 

Every term in one polynomial multiplies every term in other polynomial. 

(2x + 5) × (4x – 3) 
= 2x (4x – 3) + 5(4x – 3) 
= 2x⋅4x – 2x⋅3 + 5⋅4x –5⋅3
= 8x² − 6x + 20x −15
= 8x²  + 14x −15

In multiplication of polynomials with polynomials, we should always look for like terms, if any, and combine them.  

(a + b + c)(a + b – c)
= a(a + b – c) + b(a + b – c) + c(a + b – c)
= aa + ab – ac + ba +bb – bc + ca + cb – cc
= a²+ ab – ac + ba + b² – bc + ca + cb – c²
= a²+ b² – c² + ab + ba – bc + cb – ac+ ca 
= a²+ b² – c² + 2ab

Algebraic Expression
	Equation ➤