ALGEBRAIC EXPRESSIONS

An algebraic expression is a mathematical phrase having one or more algebraic terms including variables, constants and operating symbols (such as plus and minus signs).

While constructing an algebraic expression, we use mathematical operators like addition, subtraction, multiplication and division to combine variables and constants.

Consider the expression 2x + 3 , which is obtained by multiplying the variable x with the constant 2 and then adding the constant 3 to the product.

This expression is binomial as it contains two terms. The term 2x is a variable term and 3 is a constant term. 2 is the co-efficient of x.

Consider the algebraic expression 4x² + 5x + 7xy + 100. Vote that 4x² consists of constant 4 and variable 4x². What is the constant in the term 7xy? Is there a variable in the last term of the expression?

The ‘number parts’ of the terms with variables are coefficients. In 4x² + 5x + 7xy + 100, the coefficient of the first term is 4. What is the coefficient of the second term? It is 5. The coefficient of the xy term is 7.

The terms with same variables are called like terms. For example, –7x, 2x and 5x are like terms. But, term with different variables are called unlike terms. For example, -2x and 7y are unlike terms.

We can add or subtract like terms only. We know that

2x + 5x = 7x

But, when we add unlike terms, it results in new expression. For example, 2x and 5y are unlike terms, thus resulting in new expression 2x + 5y .

Main Concepts and Results

• Algebraic expression is formed from variables and constants using different operations.

• Expressions are made up of terms.

• A term is the product of factors. Factors may be numerical as well as algebraic (literal).

• Coefficient is the numerical factor in a term. Sometimes, any factor in a term is called the coefficient of the remaining part of the term.

• The terms having the same algebraic factors are called like terms.

• The terms having different algebraic factors are called unlike terms.

• Expression with one term is called a 'Monomial’.

• Expression with two unlike terms is called a 'Binomial’.

• Expression with three unlike terms is called a 'Trinomial’.

• In general, an expression with one or more than one term (with nonnegative integral exponents of the variables) is called a ‘Polynomial’.

• The sum (or difference) of two like terms is a like term with coefficient equal to the sum (or difference) of coefficients of the two like terms.

• When we add (or subtract) two algebraic expressions, the like terms are added (or subtracted) and the unlike terms are written as they are.

• To find the value of an expression, we substitute the values of the variables in the expression and then simplify.

• Rules and formulas in mathematics are written in a concise and general form using algebraic expressions.

Questions and Answers

Question 1 :

What is an algebraic expression with two terms called?

Answer :

Binomial

Question 2 :

Is every polynomial a monomial?

Answer :

No, the polynomial with only one term is monomial.

Question 3 :

What is trinomial? Give an example.

Answer :

An algebraic expression with exactly three terms is called trinomial.

Example :

x² - 5x + 6

Question 4 :

Is the value of a variable fixed?

Answer :

No, a variable may take any value. So, the value of a variable is not fixed.

Question 5 :

Twice the sum of length x and width y of a rectangle is the perimeter of a rectangle. Write the expression for perimeter.

Answer :

Perimeter of rectangle :

= 2(length + width)

= 2(x + y)

= 2x + 2y

Question 6 :

Identify the term containing x² in the following expression

x³ + 3x²y + 3xy² + y³

and write its coefficient.

Answer :

Term containing x³ = 3x²y

Coefficient of x² = 3y

Question 7 :

Are the following two terms like terms?

3x²y and 5x²y

Answer :

Yes, the given two terms are like terms. Because, both the terms contain the same variables x and y and each variable has the same exponent in both the terms.

Question 8 :

Are the following two terms like terms?

5x³y² and 7x³y⁵

Answer :

No, the given two terms are unlike terms. Even though both the terms contain the same variables x and y, the variable y does not have the same exponent in both the terms.

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