Multiplication of Algebraic Expressions

Multiplication of algebraic expressions
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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.
(\because a^ma^n = a^{m+n})

3x^2y\cdot 2xy^3

=3\cdot x^2 \cdot y \cdot 2 \cdot x \cdot y^3

=3\cdot 2 \cdot x^2 \cdot x \cdot y \cdot y^3

= 6x^{2+1}y^{1+3}

= 6x^3y^4

ℹ An algebraic term itself is a product of a constant and variables

Multiplying two or more monomials

xy\times yz\times zx
= x\cdot x\cdot y\cdot y\cdot z\cdot z
= x^{1+1}y^{1+1}z^{1+1}
= x^2y^2z^2

a\times (-a)^2\times a^3
= a\times a^2\times a^3
= a^{1+2+3}
= a^6

2\times 4y\times 8y^3\times 16y^3
= 2\times 4\times 8\times 16\times y^{1+3+3}
= 1024y^7

● 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\pm c) = ab \pm ac

2x\times (3x + 5xy)
= 2x\cdot 3x + 2x\cdot 5xy
= 6x^2 + 10x^2y

a^2 (2ab-5c)
= a^2\cdot 2ab-a^2\cdot 5c
= 2a^3b-5a^2c

(4p^2+ 5p + 7) \times 3p
= 4p^2 \times 3p + 5p \times 3p + 7\times 3p
= 12p^3+ 15p^2+ 21p

Multiplying polynomials

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

(2x + 5) \times (4x-3)
= 2x (4x-3) + 5(4x-3)
= 2x\cdot 4x - 2x\cdot 3 + 5\cdot 4x - 5\cdot 3
= 8x^2 - 6x + 20x - 15
= 8x^2 + 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
= a^2+ ab - ac + ba + b^2 - bc + ca + cb - c^2
= a^2+ b^2 - c^2 + ab + ba - bc + cb - ac+ ca
= a^2+ b^2 - c^2 + 2ab

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