ICSE Class 7 Maths Chapter 13 Fundamental Concepts

Unit 3 - Algebra

Chapter 13: Fundamental Concepts (Including Fundamental Operations)

13.1 Elementary Treatment

1. Constants and Variables

In Arithmetic, we use digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) each of which has a fixed value and so these digits and the numbers formed by these digits are called constants, whereas in Algebra, we use letters of English alphabet which can be assigned any value according to the requirement. So the letters used in Algebra are called variables.

(i) A combination of two or more than two constants is always a constant.

e.g. 3 is a constant and 8 is also a constant, therefore each of 3 + 8, 3 - 8, 8 - 3, 8 ÷ 3, 8 ÷ 3, 3 × 8, 38, 83, etc., is also a constant.

Similarly, 5 is a constant, 3 is a constant and 2 is also a constant; so each of 5 × 3 ÷ 2, 3 ÷ 5 × 2, 3 + 5 × 2, 352, 5 × 3 - 2, 532, etc., is also a constant.

(ii) A combination of two or more variables is always a variable.

e.g. x, y and z are variables and so each of x + y - z, x - yz, x - y + z, x + y × z, etc., is also a variable.

(iii) A combination of one or more constants and one or more variables is also a variable.

e.g. Each of 6 + x, x - 8, 10z, 8x ÷ 3y, 2x - 3y + 4z, etc., is a variable.

2. Term

A term is a number (constant), a variable or a combination (product or quotient) of numbers and variables.

e.g. \(5, 3a, x, ax, xy, -4xy, \frac{4x}{3y}, \frac{8pq}{7a}, \frac{8}{z}, \frac{5ax}{12}\), etc.

3. Algebraic Expression

An algebraic expression is a collection of one or more terms, which are separated from each other by addition (+) or subtraction (-) sign(s).

e.g. \(4x, 3xy - 7, a + 2b, 5x - 7y, 2x^2 + 5xy + 3, ax - by - cz\), etc.

Only plus (+) and minus (-) signs separate the terms, whereas the product (×) and division (÷) do not separate the terms.

For example:

The expression \(3x + 4y\) has two terms, the expression \(3x - 4y\) also has two terms but each of \(3x × 4y\) and \(3x ÷ 4y\) has only one term.

4. Types Of Algebraic Expressions

An expression of the type \(\frac{4}{x}\) does not form a monomial unless x is not equal to zero (0).

Reason: Since x is a variable, it can take any value. If it takes the value zero i.e., if \(x = 0\), the expression \(\frac{4}{x}\) is equal to \(\frac{4}{0}\), which is not defined.

Similarly, \(\frac{7}{y}\) is not a monomial unless \(y ≠ 0\),

\(\frac{15}{xy}\) is not a monomial unless \(x ≠ 0\) and \(y ≠ 0\).

5. Product

When two or more quantities (constants, variables or both) are multiplied together, the result is called their product.

For example:

(i) 4ay is the product of 4, a and y.

(iii) 8b is the product of 8 and b and so on.

6. Factor

Each of the quantities (constants or variables) multiplied together to form a term is called a factor of the term.

For example:

(i) 3, a and y are the factors of the term 3ay.

(ii) 2, a and b are the factors of the term 2ab and so on.

Infact, factor of a quantity is each and every constant, variable, combination of constant and variable, etc., by which the given quantity is completely divisible.

For example:

Quantity 3ay is completely divisible by each of 1, 3, a, y, 3a, 3y, ay and 3ay, so the factors of 3ay are: 1, 3, a, y, 3a, 3y, ay and 3ay.

Similarly, the factors of 2ab are: 1, 2, a, b, 2a, 2b, ab and 2ab.

7. Co-efficient

In a monomial, any factor or group of factors of a term is called the coefficient of the remaining part of the monomial.

For example:

In 5xyz, 5 is the co-efficient of xyz, x is the co-efficient of 5yz, y is the coefficient of 5xz, 5x is the co-efficient of yz, xy is the co-efficient of 5z and so on.

If a factor is a numerical quantity (i.e., constant), it is called numerical co-efficient, while the factor involving letter(s) is called the literal co-efficient. Thus in 5xyz, 5 is numerical co-efficient and each of x, y, z, 5x, 5y, 5z, xy, yz, xz, 5xy, 5yz, 5xz, and 5xyz are the literal co-efficients.

8. Degree Of A Monomial

The degree of a monomial is the exponent of its variable or the sum of the exponents of its variables.

For example:

(i) The degree of \(4x^2 = 2\) [Since, exponent of \(x^2\) is 2]

(ii) The degree of \(7x = 1\) [Since, \(x = x^1\)]

(iii) The degree of \(8x^2y^3 = 2 + 3 = 5\) [Sum of the exponents of the variables x and y]

(iv) The degree of \(\frac{2}{7}xy^4 = 1 + 4 = 5\)

(v) The degree of \(2 = 0\) [Since, it has no variable]

9. Degree Of A Polynomial

The degree of a polynomial is the degree of its highest degree term.

e.g. (i) In expression \(5x^4 + 7x^3y^2 + 2xy^2\), the degree of term \(5x^4 = 4\), the degree of term \(7x^3y^2 = 3 + 2 = 5\) and the degree of term \(2xy^2 = 1 + 2 = 3.\) Since, the highest degree term is \(7x^3y^2\) and its degree is 5, therefore, degree of the given polynomial is 5.

(ii) The degree of polynomial \(x^5 - x^2y^4 + x^3y\) is \(2 + 4 = 6\).

10. Like And Unlike Terms

Terms having the same literal coefficients or alphabetic letters are called like terms, whereas the terms with different literal co-efficients are called unlike terms.

For example:

(i) 5x and 8x are like terms, whereas 5x and 8y are unlike terms.

(ii) \(7x^2\) and \(2x^2\) are like terms, whereas \(7x^2\) and 2x are unlike terms.

(iii) \(3xy^2\) and \(4xy^2\) are like terms, whereas \(3xy^2\) and \(4x^2y\) are unlike terms.

Example 1

For algebraic expression \(5 - 8xy + 6x^2y^3 + 5xy^2 - 8x^3y^4\) find:

(i) number of terms

(ii) degree of the expression

(iii) coefficient of \(x^3\) in \(-8x^3y^4\)

(iv) coefficient of x in \(6x^2y^3\)

(v) constant term

(vi) literal coefficient of \(- 8xy\)

(vii) coefficient of \(x^2\) in \(-8x^3y^4\)

(viii) all the factors of \(5xy^2\)

Solution

(i) No. of terms = 5 (Ans.)

(ii) Degree of the expression = 3 + 4 = 7 (Ans.)

(iii) Coefficient of \(x^3\) in \(- 8x^3y^4 = - 8y^4\) (Ans.)

(iv) Coefficient of x in \(6x^2y^3 = 6xy^3\) (Ans.)

(v) Constant term = 5 (Ans.)

(vi) Literal coefficient of \(- 8xy = -8\) (Ans.)

(vii) Coefficient of \(x^2\) in \(-8x^3y^4 = -8xy^4\) (Ans.)

(viii) All the factors of \(5xy^2 = 1, 5, x, y, xy, y^2, xy^2, 5x, 5xy, 5y, 5y^2\) and \(5xy^2\) (Ans.)

Teacher's Note

Understanding constants and variables is like learning the difference between fixed rules (like gravity) and flexible situations (like choosing clothes) - algebra lets us work with patterns that apply in many different scenarios.

Exercise 13(A)

1. Fill in the blanks:

(i) 8 is a .............., x is a ................. and 8x is a ...............

(ii) y is a .............., 15 is a ............... and y + 15 is a ...............

(iii) 7x is a .............. y is a ............... and 7xy is a ...............

(iv) 6x + 2y + 7xy is an ...................... with ............... terms.

(v) 6x + 2y - 7xy is an ...................... with ............... terms.

(vi) 6x - 2y × 7xy is an ...................... with ............... terms.

(vii) Every binomial is a ...............

(viii) Every trinomial is a ...............

(ix) A trinomial has .............. terms, a polynomial has ...................... terms and a multinomial has ...................... terms.

(x) 7xyz is product of ...............

(xi) All possible factors of 7xyz are ...............

(xii) The degree of \(x + y^2\) is .............. and the degree of \(xy^2\) is ...............

(xiii) The degree of \(8x^5 + 7xy^3 - 6xy^8\) is ...............

(xiv) 5x and 5y are .............. terms, whereas \(5x^2y^3\) and \(28x^2y^3\) are .............. terms.

(xv) The degrees of two like terms are always ...............

(xvi) The degrees of two unlike terms can be .............. as well as ...............

2. Separate constant terms and variable terms from the following:

\(8, x, 6xy, 6 + x, -5xy^2, 15az^2, \frac{32z}{xy}, \frac{y^2}{3x}\)

3. For each expression, given below, state whether it is a monomial, binomial or trinomial:

(i) \(2x ÷ 15\) | (ii) \(ax + 9\) | (iii) \(3x^2 × 5x\)

(iv) \(5 + 2a - 3b\) | (v) \(2y - \frac{7}{3}z ÷ x\) | (vi) \(3p × q ÷ z\)

(vii) \(12z ÷ 5x + 4\) | (viii) \(12 - 5z - 4\) | (ix) \(a^3 - 3ab^2 × c\)

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