Multiplication of Algebraic Expression & Solved Examples

Multiplication of algebraic expressions is a process of multiplying two given expressions consisting of variables and constants. An algebraic expression is an expression that is built by a mixture of integer constants and variables. Multiplication of algebraic expressions can be performed on algebraic expressions in a similar way as they can be performed on two whole numbers or fractions. Multiplication of two algebraic expressions or variable expressions involves the product of two expressions which are combined with the help of arithmetic operations such as addition, subtraction, multiplication, division, and also constants, variables, terms, and coefficients. The multiplication of algebraic expression is divided into three types: monomial, binomial or polynomial.

Key Takeaways: Algebra, Monomial, Binomial, Polynomial, Like term, Unlike term, Law of Distribution, Algebraic identities, Coefficient, Variable

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Algebraic Expression

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An algebraic expression is the combination of the variables and the consonants linked by the fundamental operators like addition (+), subtraction (-), multiplication (x), etc.

Algebraic Expression

Examples of algebraic expressions are:

• x+3y, 3y-8, x+4, 3x², 5xy + 9, etc.

Expressions are formed from variables as well as consonants. The expression 4x-3 is formed from the variable ‘x’ and consonants ‘4’ and ‘3’. We know that the value of x can be anything. It may be 2, 5, -3 etc. The value of the expression varies with the value chosen for the variables it contains. Thus, as x takes on different values, the value of 4x-3 keeps changing.

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Multiplication Of Algebraic Expression Formula

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We use distinct algebraic identities for the multiplication of algebraic expressions to make the complicated problem easy. The list given below shows some of the helpful formulas and the expansion to remember while processing multiplication of algebraic expressions.

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (a + b) (a – b) = (a² – b²)

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Method of Multiplication Of Algebraic Expression

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The outcome of the multiplication of algebraic expression is gained by multiplying each term of the polynomial by the other and then taking the algebraic sum of these products. There are various types of multiplication of algebraic expressions such as:

1. Monomial by monomial / binomial
2. Binomial by binomial
3. Polynomial by monomial/binomial

In any case, we can multiply each term of the first polynomial with each term of the second.

• Multiplication Of Two Monomials

An algebraic expression is a monomial when it only consists of one term like 6ab. Monomials usually consist of variables, numbers, or multiple numbers and/or variables that are taken as products together.

Product of two monomials = Numerical coefficients x Variable parts

For example,

Find the product of 7ab and – (3a²b4)

Solution: 7ab – (3a²b⁴)

= 7 × -3 × ab × a²b⁴

= -21 × a¹⁺² × b¹⁺⁴

= -21a³b⁵.

Multiplication Of A Polynomial By A Monomial

An algebraic expression is called a polynomial when it contains variables, coefficients, that involve only the operations of addition, subtraction, etc and non-negative integer exponentiation of variables.

Multiply each term of the polynomial by the monomial,

Using the distributive law: a (b + c) = (a b) + (a c)

For example,

• Find the following product: 6a²b² (4a² – 5ab + 6b²)

Solution: 6a²b² x (4a² – 5ab + 6b²)

= (6a²b²) (4a²) + (6a²b²) (– 5ab) + (6a²b²) (6b²)

= 24a⁴b² – 30a³b³ + 36a²b⁴.

Multiplication Of Two Binomials

An algebraic expression is considered binomial when it is made of the sum or difference of two terms. We multiply two binomials by

Using the law of distribution of multiplication twice.

= (a+ b) (c + d)

= a (c + d) + b (c + d)

= (a c) + (a d) + (b c) + (b d)

= ac + ad + bc +bd.

Note: This process is also termed as horizontal multiplication method.

Example: Multiply (4a + 5b) and (5a – 6b)

Solution: I. Horizontal multiplication method:

(4a + 5b) and (5a – 6b)

= 4a (5a – 6b) + 5b (5a – 6b)

= (4a 5a) – (4a 6b) + (5b 5a) – (5b 6b)

= 20a² – 24ab + 25ab -30b²

= 20a² + ab -30b².

1. Vertical multiplication method:

(4x + 5y) and (5x – 6y)

= (4x + 5y)

(5x – 6y)

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 20x² + 25xy ---- multiply by 5x

-24xy – 30y²-----multiply by -6y

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20x² + xy -30y². --- add the above terms.

Multiplication By A Polynomial

An expression containing more than one term with a non-zero coefficient with a variable having non-negative exponents is termed as a polynomial.

Example: Multiply (6x²-5x + 8) with (2x -3)

Solution:

(6x²- 5x + 8)

(2x -3)

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2x (6x²) – 2x (5x) + 2x (8) ---- multiply by 2x

– 3 (6x²) + 3x (5x) + 3 (8) -----multiply by 3

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= 12x³ -10x² +16x -18x² + 15x² + 24

= 12x³ - 13x² + 16x +24. ----- added the above terms.

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Applications Of Multiplication Of Algebraic Expression

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The multiplication of algebraic expression helps in solving problems in the following way:

• We can add or subtract terms only if they are like terms.
• For division or multiplication of algebraic expression, the terms can be either like or unlike terms.
• Multiplication of algebraic expressions can be done with the assistance of algebraic identities.

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Things To Remember

• An algebraic expression signifies a series of operations that can also be expressed in words. 
• The sign of multiplication is often ignored in algebraic expressions: we usually write 2x instead of 2 * x and 4 (a +b) instead of 4* (a + b). 
• It is accepted worldwide that, in expressions that do not contain brackets, addition, and subtraction should be performed as they seem from left to right in the expression.
• An algebraic expression is considered to be binomial when it is formed of the sum or difference of two terms. 
• An expression with more than one term with a non-zero coefficient with a variable having non-negative exponents is a polynomial.

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