Algebraic Expressions and Identities

Multiplication of algebraic expressions

Polynomial is a special kind of algebraic expression. In a polynomial, all variables are raised to only whole numbers (0, 1, 2, 3, 4,...) powers.

An algebraic expression which contains only one term is called a monomial.

Example: 3xyz, 4m2, −17a13 and r7 are monomials.

An algebraic expression which contains two terms is called a binomial.

Example: x−y4, 6a+17b, 3m−15 and 34u2 v+4u4 v 3 are binomials.

An algebraic expression which contains three terms is called a trinomial.

Example: a+b−c, 2(x2+5y+z), m3+15n2−37m and 2p2 q−5pq 2−29s  are trinomials.

An algebraic expression which contains one or more than one is called a polynomial.

Example: 2p−q+3r 3−7/2s, m 4+n 3m−6m 2 +14m 2n 2+56 and x 3+y 3−3xyz are polynomials.

Introduction to product of algebraic expression

On the Children day celebration, Madhu arranged seating for children.

Situation 1: There are 'x' unknown number of rows and 'y' unknown number of columns. The number of chairs can be arranged will be expressed as the product of two variables.

That is x×y.

Thus, the number of chairs Madhu arranged for seating is x×y.

Situation 2: Suppose there are 'x' unknown number of rows and y2+2y number of columns. The number of chairs can be arranged will be expressed as the product of two polynomials.

That is x×(y2+2y).

Thus, the number of chairs Madhu arranged for seating is x×(y2+2y).

Types of multiplication of algebraic expression

1. Look for the sign of the terms. That is, if both the terms have like signs (+,+) (or) (−,−) then the result will have + sign. If both the terms have unlike signs (+,−) then the result will have a −ve sign.
2. Multiply the corresponding coefficient of the terms.
3. Multiply the variables using the law of exponents.

Product of monomial by a monomial

Multiplying  two monomial:

Let's take two monomials 6 and y.

We can multiply and write them as 6×y=6y=y+y+y+y+y+y (or) adding y for 6 times.

This is easy to multiply because one monomial is a variable and one monomial is a number.

If we have 2 monomials with variables and numbers, then the product is the result of (product of coefficients of monomials) × (product of variables of monomials).

ExampleTo multiply 4x and 5y.

The coefficients are 4 and 5, and the variables are x and y.

4x×5y= (4×5)×(x×y) =20xy.

To multiply monomials of higher degrees, find the product of coefficients of monomials and then use the law of exponents to add the power of similar variables.

The useful law of exponents are as follows:

1. am×an=am+n
2. (am)n=amn

If we have both the monomials as the same variables, then use the law of exponents, am×an=am+n to find the product. For example, y2×y2=y2+2=y4. This is not feasible in case of different variables.

Example: If we have to multiply monomials of higher degrees, such as 4x 3and5xy4, it will be 20x 4y 4, find the product of coefficients of monomials =20 and then use the law of exponents to add the power of similar variables, am×an=am+n  to find x3×x1=x4 and then y4. Hence, we have 20x 4y 4.

Multiplying more than two monomials:

To multiply more than two monomials, extend the idea of multiplying two monomials to more than monomials.

Example: Find the volume of a cuboid whose sides are −2x 2,4xy 3and −5y 2 z 3.

This can be understood with the help of an example having three monomials.

−2x2,4xy3 and −5y2 z3

Let us multiply the three monomials.

=(−2×4×−5)×(x2×xy3×y 2z 3)

=(−×+×−)2×4×5 ((x2×x)×(y3×y2)×(z3))

=40(x 3×y 5×z 3)

=40x 3y 5z3

Product of monomial by a polynomial

distributive proportion.

If a is a constant, x and y are variables, then a(x+y)=ax+ay.

Suppose there will be x number of bags and a bag contains 3 cupcakes of 'p' packs, 7 chocolates of 'q' packs and 5 cookies of 'r' packs. The total number of items can be identified by adding the number of items in the bag and product with the number of bags.

This can be written as x(3p+7q+5r).

Applying the distributive property,

=3px+7qx+5rx.

Product of polynomial by a polynomial

In this multiplication, we have to multiply each of the terms in the first polynomial by each of the terms in the second polynomial.

Consider a trinomial a+b−c and a polynomial 2a−3b+5c.

Let us multiply the trinomial a+b−c by a polynomial 2a−3b+5c.

Note we have to multiply each of the terms in the first polynomial a+b−c by each of the terms in the second polynomial 2a−3b+5c.

(a+b−c)×(2a−3b+5c)

=(a×2a)+(a×−3b)+(a×5c)+(b×2a)+(b×−3b)+(b×5c)+(−c×2a)+(−c×−3b)+(−c×5c)

=2a 2−3ab+5ac+2ab−3b2+5bc−2ac+3bc−5c2

=2a2−3b2−5c2−ab+8bc+3ac.

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