ICSE Class 8 Maths Chapter 15 Simplification of Algebraic Fractions

Chapter 15: Simplification Of Algebraic Fractions

You have worked on numerical fractions like \(\frac{3}{12}, \frac{5}{7}, -\frac{4}{9}, 1\frac{3}{4}\) and algebraic fractions like \(\frac{x}{3}, \frac{2y+3}{7}, \frac{x+2y-1}{5}, \frac{5}{2x-1}\) etc. Working with algebraic fractions often involves the concept of H.C.F. and L.C.M. of polynomials (with integral coefficients). In this chapter, we shall introduce the idea of H.C.F. and L.C.M. of polynomials and strengthen the concept of simplification of algebraic expressions.

H.C.F. And L.C.M. Of Polynomials

H.C.F. of two or more polynomials (with integral coefficients) is the highest common factor of the given polynomials.

In particular,

H.C.F. of two or more monomials = (H.C.F. of their numerical coefficients) \(\times\) (H.C.F. of their literal coefficients)

H.C.F. of literal coefficients = product of each common literal raised to the lowest power.

Remark: Here it is understood that the numerical coefficients of the monomials (under consideration) are integers and the powers of the literals involved in the monomials are positive integers.

L.C.M. of two or more polynomials (with integral coefficients) is the smallest common multiple of the given polynomials.

In particular,

L.C.M. of two or more monomials = (L.C.M. of their numerical coefficients) \(\times\) (L.C.M. of their literal coefficients)

L.C.M. of literal coefficients = product of each literal raised to the highest power.

Example 1.

Find H.C.F. and L.C.M. of the monomials \(24x^4y^3\) and \(32x^2y^4z\).

Solution.

H.C.F. of numerical coefficients = H.C.F. of 24 and 32 = 8

H.C.F. of literal coefficients = product of each common literal raised to the lowest power

\[= x^2y^3\]

\(\therefore\) H.C.F. of the given monomials = \(8 \times x^2y^3 = 8x^2y^3\)

L.C.M. of numerical coefficients = L.C.M. of 24 and 32 = 96

L.C.M. of literal coefficients = product of each literal raised to the highest power

\[= x^4y^4z\]

\(\therefore\) L.C.M. of the given monomials = \(96 \times x^4y^4z = 96x^4y^4z\).

Example 2.

Find H.C.F. and L.C.M. of the monomials \(12a^3b^2, 16ab^3c^2\) and \(20a^2b^2c\).

Solution.

H.C.F. of numerical coefficients = H.C.F. of 12, 16 and 20 = 4

H.C.F. of literal coefficients = product of each common literal raised to the lowest power

\[= ab^2\]

\(\therefore\) H.C.F. of the given monomials = \(4 \times ab^2 = 4ab^2\)

L.C.M. of numerical coefficients = L.C.M. of 12, 16 and 20 = 240

L.C.M. of literal coefficients = product of each literal raised to the highest power

\[= a^3b^3c^2\]

\(\therefore\) L.C.M. of the given monomials = \(240 \times a^3b^3c^2 = 240a^3b^3c^2\).

Method To Find H.C.F. And L.C.M. Of Polynomials

To find H.C.F. and L.C.M. of given polynomials - factorise each polynomial, then

(i) H.C.F. = (H.C.F. of numerical coefficients) \(\times\) (each common factor raised to the lowest power)

(ii) L.C.M. = (L.C.M. of numerical coefficients) \(\times\) (each factor raised to the highest power)

Example 3.

Find the H.C.F. and L.C.M. of the polynomials \(4x^2 + 4xy, 6x^2 - 6y^2\).

Solution.

\(4x^2 + 4xy = 4x(x + y)\)

and \(6x^2 - 6y^2 = 6(x^2 - y^2) = 6(x + y)(x - y)\)

Factorise given polynomials

H.C.F. of numerical coefficients = H.C.F. of 4 and 6 = 2

\(\therefore\) H.C.F. of the given polynomials = \(2(x + y)\)

L.C.M. of numerical coefficients = L.C.M. of 4 and 6 = 12

\(\therefore\) L.C.M. of the given polynomials = \(12x(x + y)(x - y)\).

Example 4.

Find the H.C.F. and L.C.M. of \(5a^2 + 15a, a^2 + 6a + 9\) and \(2a^2 - 18\).

Solution.

Factorising the given polynomials, we get

\(5a^2 + 15a = 5a(a + 3),\)

\(a^2 + 6a + 9 = (a)^2 + 2 \times a \times 3 + (3)^2 = (a + 3)^2\)

and \(2a^2 - 18 = 2(a^2 - 9) = 2(a + 3)(a - 3)\)

H.C.F. of numerical coefficients = H.C.F. of 5, 1 and 2 = 1

\(\therefore\) H.C.F. of the given polynomials = \(1 \times (a + 3) = a + 3\)

L.C.M. of numerical coefficients = L.C.M. of 5, 1 and 2 = 10

\(\therefore\) L.C.M. of the given polynomials = \(10a(a + 3)^2(a - 3)\).

Example 5.

Find the H.C.F. and L.C.M. of \(12x^2 - 75, 4x^2 - 20x + 25\) and \(6x^2 - 13x - 5\).

Solution.

Factorising the given polynomials, we get

\(12x^2 - 75 = 3(4x^2 - 25) = 3(2x + 5)(2x - 5),\)

\(4x^2 - 20x + 25 = (2x)^2 - 2 \times 2x \times 5 + (5)^2 = (2x - 5)^2\)

and \(6x^2 - 13x - 5 = 6x^2 + 2x - 15x - 5\)

We need two integers whose sum is -13 and product is \(6 \times (-5)\) i.e. -30.

By trial, \(2 + (-15) = -13\) and \(2 \times (-15) = -30\)

\(= 2x(3x + 1) - 5(3x + 1)\)

\(= (3x + 1)(2x - 5).\)

H.C.F. of numerical coefficients = H.C.F. of 3, 1 and 1 = 1

\(\therefore\) H.C.F. of the given polynomials = \(1 \times (2x - 5)\)

\(= 2x - 5\)

L.C.M. of numerical coefficients = L.C.M. of 3, 1 and 1 = 3

\(\therefore\) L.C.M. of the given polynomials = \(3(2x + 5)(2x - 5)^2(3x + 1)\).

Teacher's Note

Understanding H.C.F. and L.C.M. of polynomials helps students grasp how common factors work, similar to finding the greatest common divisor when sharing items equally among groups in real life.

Exercise 15.1

Find the H.C.F and L.C.M. of the following (1 to 4) monomials:

1. (i) \(6x^2y\) and \(4xy^2\) (ii) \(12a^3b^2\) and \(18a^2b^5\)

2. (i) \(12x^3y^3\) and \(28x^2y^2z\) (ii) \(48a^2bc\) and \(56ab^2c^3d\)

3. (i) \(2m^3n^2, 3mn^3\) and \(5m^2n\) (ii) \(6a^2b^2, 15ab^4\) and \(12a^4b^2\)

4. (i) \(9p^3q^2r, 36pq^3r^3\) and \(12p^2q^3r^3\) (ii) \(10x^3y^2z^4, 15x^2y^3\) and \(20yz^2\)

Find the H.C.F and L.C.M. of the following (5 to 8) polynomials:

5. (i) \(x^2 + 3xy\) and \(x^2 - 9y^2\) (ii) \(9x^2 - 16y^2\) and \(15x^2 - 20xy\)

6. (i) \(4a^2 - 25\) and \(4a^2 - 20a + 25\) (ii) \(6x^2 + 12xy\) and \(4x^2 - 16y^2\)

7. (i) \(x^2 + 6x + 9\) and \(2x^2 + 7x + 3\) (ii) \(4a^2 + 4a + 1\) and \(6a^2 + 7a + 2\)

8. (i) \(3x^2 + 6x, x^2 + 5x + 6\) and \(2x^2 + 8x + 8\)

(ii) \(4x^4 - 36, 2x^3 - 12x + 18\) and \(2x^2 + x - 21\)

Algebraic Fractions

Fractions involving polynomials either in numerator or denominator (or both) are called algebraic fractions.

For example:

(i) \(\frac{2x}{7}, \frac{3x-2}{5}, \frac{x+2y-7}{11}, \frac{2x^4+5}{3}\) etc. are algebraic fractions with integral denominators.

(ii) \(\frac{a}{b}, -\frac{3}{x}, \frac{1}{x+2}, \frac{2x+3}{x-5}, \frac{x^2-1}{2x+1}, \frac{3x^2+5}{x^2+4}\) etc. are algebraic fractions involving variables in the denominators.

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