Algebraic expressions: Overview, Questions, Preparation

An algebraic expression comprises variables and constants connected with mathematical operators like ‘+’ and ‘-”. Each element in the expression is referred to as a term. Variables, coefficients, and constants together form an expression.

Types of Algebraic Expressions

Algebraic expressions can be categorised into monomial, binomial, and polynomial expressions. Collectively, they are known as polynomials. 

• Polynomials: A polynomial cannot have negative power, fractions, or square roots on its variables, and the denominators cannot have any variables. For instance, 1/x² is not a polynomial.
• Monomials - Expressions that have only one term. For example, 4xy, 8y, 4x², etc.
• Binomials - Expressions that have two terms. 3xy+7, 3x³+y, etc.
• Trinomials - Expressions that have three terms. For example, x²+7x+ y. 

Addition & Subtraction of Algebraic Expressions

During addition or subtraction of two expressions, the like-terms are added or subtracted with each other, respectively. 

The numerical coefficient will be equal to the sum of both the terms’ numerical coefficients while adding two expressions. During subtraction, the numerical coefficient will be equal to the difference of the numerical coefficients.

For instance, if A = 3x²+4y and B = 13y-x² , then A+B = 2x²+17y and A-B = 4x²-9y.

Like & Unlike Terms

Terms having similar variables and powers are like-terms, whereas those having different variables and powers are known as unlike-terms. For example, 4x² and x² are like-terms and 3x and 4y are unlike-terms.

Finding Values of an Expression

While finding the value of an expression, the values of the variables are substituted in the expressions. For example, if x =1 in 2x³+16x+16, then the value of the expression will be equal to 2+16+16 = 34.

Methods of Finding the Square Root of an Expression

An algebraic expression’s square root can be obtained using two methods, viz. division method and factoring method. The division method is a long process where each term’s square root is calculated using the variable and coefficient that reach the nearest to the variable with the highest power.

The factoring method follows the factor theorem as per which p is the factor of f(x) if it is fully divisible with p, i.e., if the remainder becomes zero after dividing it with p. This means that x-p or f(p) = 0 will be the factors of the term f(x). 

1. If a = 3 and b = 4, then solve the algebraic expression 58-3a+7b.

Solution.
58- 3(3)+7(4) = 58 - 9 + 28 = 77

2. Rohan eats the same toffee three times a day. Give the algebraic expression for one week of his toffee eating habit.

Solution.
Suppose that Rohan eats the toffee ‘p’. Therefore, in one day he will eat 3p toffees, and in one week he will eat 3p x 7 toffees.

3. Find the value of x in x³ + 27y + 9, if the total value of the expression is 72 and y = 2

Solution.
As per the above statement, x³+27y+9 = 72
Therefore, x³ + 27y = 63
x^3 = 63 - 54
x^3 = 9
Therefore, x = 2.08.

Q: Can a single-digit number be referred to as an expression?

Q: How to define a variable in an algebraic expression?

Q: Is every algebraic expression a polynomial?

Q: Are algebraic expressions the same as algebraic equations?

Q: How to find the degree of a polynomial?