ICSE Class 8 Maths Chapter 10 Fundamental Concepts and Operations on Algebraic Expressions

Chapter 10: Fundamental Concepts And Operations On Algebraic Expressions

In previous classes, you studied the fundamental concepts of algebra and the operations on algebraic expressions. In this chapter, we shall review these topics and solve a few tougher problems.

Fundamental Concepts

In algebra, we use two types of symbols - constants and variables (literals).

Constant. A symbol which has a fixed value is called a constant.

For example, each of 7, -3, \(\frac{2}{5}\), \(-\frac{7}{3}\), \(\sqrt{2}\), \(2 - \sqrt{3}\), \(\pi\) etc. is a constant.

Variable. A symbol which can be given various numerical values is called a variable or literal.

For example, the formula for circumference of a circle is \(C = 2\pi r\), where C is the length of the circumference of the circle and \(r\) is its radius. Here 2, \(\pi\) are constants and C, r are variables (or literals).

Algebraic Expression

A collection of constants and literals (variables) connected by one or more of the operations of addition, subtraction, multiplication and division is called an algebraic expression.

The various parts of an algebraic expression separated by '+' or '-' sign are called terms of the algebraic expression. Various types of algebraic expressions are:

Monomial. An algebraic expression having only one term is called a monomial.

Binomial. An algebraic expression having two terms is called a binomial.

Trinomial. An algebraic expression having three terms is called a trinomial.

Multinomial. An algebraic expression having two or more terms is called a multinomial.

For example:

Teacher's Note

Algebraic expressions are used in daily life when calculating costs, areas, and other quantities that vary - for instance, determining the total price of multiple items at different rates, or finding the area of rooms with variable dimensions.

Remark

Multiplication and division do not separate the terms of an algebraic expression. Thus, \(-7x^3y\) is one term while \(-7x^3 + y^3\) has two terms.

Factors. Each of the quantity (constant or literal) multiplied together to form a product is called a factor of the product.

A constant factor is called a numerical factor and any factor containing only literals is called a literal factor.

In \(-7xy^2\), the numerical factor is -7 and the literal factors are x, y, \(y^2\), xy and \(xy^2\).

Constant term. The term of an algebraic expression having no literal factors is called its constant term.

In the expression \(-3x^2y^3 + \frac{5}{x} - 7\), -7 is the constant term,

while the expression \(9x^5 - 3x^2 + \frac{11}{x^3}\) has no constant term.

Coefficient. Any factor of a (non-constant) term of an algebraic expression is called the coefficient of the remaining factor of the term.

In particular, the constant part is called the numerical coefficient or simply the coefficient of the term and the remaining part is called the literal coefficient of the term.

Consider the expression \(7p^3q^2 - 5p^2q - 3p + 2\). In the term \(-5p^2q\):

the numerical coefficient = -5,

the literal coefficient = \(p^2q\),

the coefficient of \(p^2\) = -5q,

the coefficient of 5p = -pq,

the coefficient of -q = \(5p^2\) etc.

Note. When we write x, we mean 1x. Thus, if no coefficient is written before a literal, then the coefficient is always taken as 1.

Like and unlike terms. The terms having same literal coefficients are called like terms; otherwise, they are called unlike terms.

For example:

(i) \(5x^2yz, -3x^2yz, \frac{3}{5}yzx^2\) are like terms

(ii) \(7ab, -3a^2b, \frac{2}{3}ab^2\) are unlike terms.

Polynomial

An algebraic expression is called a polynomial if the powers of the variables involved in it in each term are non-negative integers.

Polynomial In One Variable

An algebraic expression containing only one variable (literal) is called a polynomial in that variable if the powers of the variable in each term are non-negative integers.

The greatest power of the variable in a polynomial is called its degree.

For example:

(i) \(7 - 3x\) is a polynomial in x of degree 1.

(ii) \(5t^2 - \frac{2}{3}t + 8\) is a polynomial in t of degree 2.

(iii) \(\frac{3}{4}p^3 - 5p^2 - \frac{2}{3}\) is a polynomial in p of degree 3.

(iv) \(9x^6 - 3x^2 + 4x - \frac{1}{7}\) is a polynomial in x of degree 6.

(v) \(5x^2 - \frac{3}{x} + 6\) is not a polynomial. Note that this expression is a trinomial.

Polynomial In Two Or More Variables

An algebraic expression containing two or more variables (literals) is called a polynomial in those variables if the powers of the variables in each term are non-negative integers. Take the sum of the powers of the variables in each term; the greatest sum is the degree of the polynomial.

For example:

(i) \(8x^3y^2 - 5x^2y^3 - \frac{2}{3}xy^3 + 7xy - \frac{1}{5}\) is a polynomial in two variables x and y.

The degrees of its terms are 3 + 2, 2 + 3, 1 + 3, 1 + 1, 0.

So, the degree of the polynomial is 5.

(ii) \(5a - 3abc + \frac{7}{3}ab^2 - 9bc^3 + 6\) is a polynomial in three variables a, b, and c.

The degrees of its terms are 1, 1 + 1 + 1, 1 + 2, 1 + 3, 0.

So, the degree of the polynomial is 4.

(iii) \(5pq^2 - \frac{3p}{q} + 7q^3 - 8\) is not a polynomial.

Linear polynomial. A polynomial of degree 1 is called a linear polynomial.

For example:

(i) \(3x + 7, 5 - 9y, 6p - \frac{2}{3}, 7a\) are all linear polynomials in one variable.

(ii) \(3x + 5y, 2p - 3q + 7, 5a - \frac{2}{3}b + 11\) are all linear polynomials in two variables.

(iii) \(2x + 3y - 7z, 2a - 3b + 5c - 4\) are linear polynomials in three variables.

Quadratic polynomial. A polynomial of degree 2 is called a quadratic polynomial.

For example:

(i) \(3x^2 - 5x + 2, 7y^2 - 9, 4p^2 + 3p\) are all quadratic polynomials in one variable.

(ii) \(3x^2 - 5xy + 7y^2, 7p^2 - 9pq + 3p - \frac{2}{3}\) are quadratic polynomials in two variables.

(iii) \(3a^2 + 2b^2 - 3c + 5ab - 4\) is a quadratic polynomial in three variables.

Cubic polynomial. A polynomial of degree 3 is called a cubic polynomial.

For example:

(i) \(5x^3 - 7x^2 + 2, 1 + 7p - 9p^2 + \frac{2}{3}p^3\) are cubic polynomials in one variable.

(ii) \(3xy^2 - 7y^3 + 2x^2 - 3xy + \frac{2}{5}y\) is a cubic polynomial in two variables.

Teacher's Note

Understanding polynomial degrees helps in fields like engineering and physics, where models of increasing complexity are needed - linear for simple relationships, quadratic for motion under gravity, and cubic for more complex phenomena.

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