ICSE Class 8 Maths Algebra Chapter 01 Fundamental Concepts and Operations

Fundamental Concepts and Operations

Fundamental Concepts

You are already familiar with the basic terms and concepts of algebra. Let us recall what you have learnt in previous classes.

Literals and Constants

A symbol, such as a, b, c, x, y and z, representing an unspecified number or member of a class of objects is called a literal or a variable.

A symbol which has a fixed value is known as a constant. Numbers such as 36, 0, \(\frac{2}{3}\), 5.6 and -2 have a specific value, so they are constants.

Operations on Literals

Since literals represent numbers, we can add, subtract, multiply and divide literals and numbers.

Addition

(i) \(x + y\) is the sum (or addition) of two literals x and y.

(ii) \(x + x = 2x, x + x + x = 3x\), etc.

(iii) \(x + 0 = x, 0 + x = x\).

Subtraction

(i) \(a - b\) means b subtracted from a.

(ii) \(a - a = 0\).

(iii) 5 subtracted from a is \(a - 5\).

Multiplication

(i) \(ab\) means \(a \times b\) or a multiplied by b.

(ii) \(5x\) means \(5 \times x\) or 5 multiplied by x.

(iii) \(1 \cdot x = x\).

Division

(i) \(a \div b\) or \(\frac{a}{b}\) means a divided by b.

(ii) \(a \div 3\) or \(\frac{a}{3}\) means a divided by 3.

(iii) \(6 \div a\) or \(\frac{6}{a}\) means 6 divided by a.

(iv) \(\frac{x}{x} = 1\).

(v) \(\frac{x}{1} = x\).

Teacher's Note

Just as we use variables in algebra to represent unknown quantities, we do the same when budgeting - using symbols to represent costs or quantities we need to track.

Laws of Operations

All the laws of addition and multiplication of numbers are valid for operations on literals.

1. \(a + b = b + a\) - (commutative law of addition)

2. \(ab = ba\) - (commutative law of multiplication)

3. \(a + (b + c) = (a + b) + c\) - (associative law of addition)

4. \(a(bc) = (ab)c\) - (associative law of multiplication)

5. \(a(b + c) = ab + ac\) - (distributive law)

Powers of a Literal

Product of a literal multiplied by itself is called a power of the literal, for example, \(x \times x = x^2\) and \(x \times x \times x = x^3\). The literal is called the base of the power and the number of times it is multiplied is called the index or exponent of the power.

Examples

(i) In \(x^2\), x is the base and 2 is the index of the power.

(ii) In \(y^3\), base = y and index of power = 3.

(iii) In \(p \times p \times p \times p \times p = p^5\), base = p and index of power = 5.

Terms

A combination of constants and variables connected only by the operations of multiplication and division is called a term. Some examples are \(4a, -3x^2, 2xy\) and \(\frac{16}{3}yz\).

Algebraic Expressions

A collection of numbers and variables connected by operations is called an algebraic expression or simply an expression. Parts of an expression connected by '+' or '-' signs are called terms of the expression. If a part has a negative sign before it, the '-' sign is included in the term.

Examples

(i) \(3x - 5y + 6z\) has three terms \(3x, -5y\) and \(6z\).

(ii) \(\frac{5}{2}ab + \frac{6}{5}bc - \frac{8}{9}cd + 7f\) has four terms \(\frac{5}{2}ab, \frac{6}{5}bc, -\frac{8}{9}cd\) and \(7f\).

If an algebraic expression has one term, it is called a monomial and if it has more than one term, it is called a multinomial. Multinomials with two and three terms are called binomials and trinomials respectively.

Factors and Coefficients

The constants and variables that are multiplied together to form a term are called factors of the term. Each factor is a coefficient of the product of the remaining factors of the term. In particular, the number or constant which appears in a term is called the numerical coefficient of the term.

Example

Consider the term \(5x^2yz\).

5 is the numerical coefficient of \(x^2yz\).

\(5x^2\) is the coefficient of \(yz\).

\(5y\) is the coefficient of \(x^2z\).

\(5yz\) is the coefficient of \(x^2\).

\(5xy\) is the coefficient of \(xz\).

Teacher's Note

Understanding coefficients is like identifying ingredients in a recipe - the numerical coefficient tells us the quantity or amount of each variable component we're working with.

Like and Unlike Terms

Terms which are alike in their variables are called like terms. Terms which are not like are called unlike terms.

Examples

(i) \(-\frac{2}{3}ab, 5ab\) are like terms.

(ii) \(\frac{5}{6}x, -\frac{5}{6}xy\) are unlike terms.

(iii) \(3x^2, -4x^3\) are unlike terms.

Degree of a Term

The highest index of power or the sum of the indices of power of the variable(s) in a term is called the degree of the term.

Examples

(i) \(-6x^2\) is of the second degree.

(ii) The degree of \(xy^2\) is \(3(= 1 + 2)\).

(iii) \(12x^8y^9z^5\) is a term of degree 22 (= 8 + 9 + 5).

Polynomial in One Variable

An algebraic expression is called a polynomial if it is a finite sum of terms which contain only non-negative integral exponents of a variable. A polynomial in one variable (say x) contains only such terms as can be expressed in the form \(ax^n\), where a is a constant and n is a non-negative integer. In other words, a polynomial in x cannot have terms, such as \(\frac{1}{x}, \frac{1}{x^2}\) and \(\frac{1}{x^3}\) because they are not of the form \(x^n\), where n is a non-negative integer.

Examples

(i) \(-3x^2 + 4x + 2\) is a polynomial in x with three terms \(-3x^2, 4x\) and 2. The term 2 is called a constant term.

(ii) \(x^2 + \frac{2}{x}\) is not a polynomial because the term \(\frac{2}{x}\) is not of the form \(ax^n\), where n is a non-negative integer.

Polynomial in Two or More Variables

A polynomial in two or more variables is a sum of terms that contain only non-negative integral exponents of those variables. For example, a polynomial in x, y and z is a sum of one or more terms of the form \(ax^m y^n z^p\), where a is any constant and m, n and p are non-negative integers.

Examples

(i) \(5xy - 2yz + 3zx\) is a polynomial in three variables x, y and z.

(ii) \(6a^2bc + 8ab^2cd + 9abc^2d^2 + 7\) is a polynomial in four variables a, b, c and d.

Teacher's Note

Polynomials help us model real-world situations - for instance, the cost of a product might be expressed as a polynomial in terms of quantity and other factors.

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