In earlier classes, we have already become familiar with what algebraic expressions
(or simply expressions) are. Examples of expressions are:You can form many more expressions. As you know expressions are formed from
variables and constants. The expression 2y – 5 is formed from the variable y and constants
2 and 5. The expression 4xy + 7 is formed from variables x and y and constants 4 and 7.
We know that, the value of y in the expression, 2y – 5, may be anything. It can be,
Number line and an expression: Consider the expression x + 5. Let us say the variable x has a position X on the number line;
Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors. The term 4x is the product of its factors 4 and x. The term 5 is made up of just one factor
Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one.
7x, 14x, –13x are like terms.
7xy and –5yx are like terms.
Observe how we do the addition. We write each expression to be added in a separate row. While doing so we write like terms one below the other, and add them,
To find the number of dots we have to multiply the expression for the number of rows by the expression for the number of columns.
Notice that all the threeproducts of monomials, 3xy,15xy, –15xy, are alsomonomials.
It is clear that we first multiply the first two monomials and then multiply the resulting monomial by the third monomial. This method can be extended to the product of any number of monomials.
Let us multiply the monomial 3x by the binomial 5y + 2,Recall that 3x and (5y + 2) represent numbers. Therefore, using the distributive law,What about a binomial × monomial? For example, (5y + 2) × 3x = ? We may use commutative law asMultiply each term of the trinomial by the monomial and add products. Observe, by using the distributive law, we are able to carry out the multiplication term by term.
9.9.1 Multiplying a binomial by a binomial
Let us multiply one binomial (2a + 3b) by another binomial, say (3a + 4b). We do this
step-by-step, as we did in earlier cases, following the distributive law of multiplication,
When we carry out term by term multiplication, we expect 2 × 2 = 4 terms to be present. But two of these are like terms, which are combined, and hence we get 3 terms. In multiplication of polynomials with polynomials, we should always look for like terms, if any, and combine them.
We shall find that for any value of a, LHS = RHS. Such an equality, true for every value of the variable in it, is called an identity. Thus, An equation is true for only certain values of the variable in it. It is not true for all values of the variable. For example, consider the equation
We shall now study three identities which are very useful in our work. These identities are obtained by multiplying a binomial by another binomial.
We shall now see how, for many problems on multiplication of binomial expressions and also of numbers, use of the identities gives a simple alternative method of solving them.