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# Division of Algebraic Expressions

we are going to learn about another basic operation ‘division’ on algebraic expressions. We know that the division is the reverse operation of multiplication.

Division of Algebraic Expressions

In the previous sessions, we have learnt how to add, subtract and multiply algebraic expressions. Now, we are going to learn about another basic operation ‘division’ on algebraic expressions. We know that the division is the reverse operation of multiplication.

Now, the cost of 10 balls at the rate of 5 each = 10 × 5

=50

whereas if we have 50 and we want to buy 10 balls then, the cost of each ball is

= 50/10 = ₹5

What we have seen above is division on numbers. But how will you divide an algebraic expression by another algebraic expression?

Of course, the same procedure has to be followed for the algebraic expressions with the help of laws of exponents.

If x is a variable and m, n are constants, then xm ÷ xn = xm n where m > n .

1. Division of a monomial by another monomial

Dividing a monomial 10 p4 by another monomial 2p3, we get

10p4 ÷ 2p3 However, to divide we can also follow laws of exponents as, Think

Are the following correct?

(i) x3/x8 = x8-3 = x5

(ii) 10m4 / 10m4 = 0

(iii) When a monomial is divided by itself, we will get 1?

Solution:

(i) x3 / x8 = x8 – 3 = x5

x3 / x8 = x3 – 8 = x5 (or) x3 / x8 = 1 / x8 – 3 = 1 / x5

(ii) 10m4 / 10m4 = 0

10m4 / 10m4 = [10/10] m4−4 = 1 m0 = 1     [ m0 = 1]

The given answer is not correct

(iii) When a monomial is divided by itself, we will get 1?

When a monomial is divided by itself, we will get 1.

Eg. x/x = x1 – 1 = x0 = 1

The given statement is correct

Example 3.6

Velu pastes ‘4xy ’ pictures in one page of his scrap book. How many pages will he need to paste 100x2 y3 pictures? (x, y are positive integers) Solution:

Total number of pictures = 100x2 y3

Pictures in one page = 4xy

Total number of pages needed = Total number of pictures / pictures in one page = 25xy2 pages

Try these

Divide

(i) 12x 3y2 by x2y (ii) 20a5b2 by 2a3b7 (iii) 28a4 c2 by 21ca2

(iv) (3 x 2 y)3 6x2y3 (v) 64m4 (n2)3 ÷ 4m2n2 (vi) (8 x 2 y 2 )3 ÷ (8x 2 y2 )2

(vii) 81 p2q4 ÷ √[81p2q4 ]

(vii) ( 4x2 y3 )0 ÷ [( x3)2]/x6

Solution: Solution:

(i) 12x3y2 / x2y = 12x3−2 y2−1  = 12x1y1 = 12 xy

(ii) −20a5b2 / 2a3b7 = −20a5 − 3 / 2b7 − 2  = −10a2 / b5

(iii) 28 a4c2 / 21ca2 = [28 / 21] a4 – 2 c2 – 1 = 4/3 a2c1 = 4/3 a2c

(iv) (3x2y)3 / 6x2y3 = 27 x6y3 / 6 x2y3  = [9/2] x6–2 = 9/2 x4

(v) 64m4(n2)3 / 4m2n2 = 64m4n6 / 4m2n2 = 16 m4−2 n6−2 = 16 m2n4

(vi) (8x2y2)3 / (8x2y2)2 = 512 x6y6 / 64 x4y4

= 8x6−4 y6−4

= 8x2y2

(vii) 81p2q4 / √81p2q4 = 81p2q4 / 9pq2 = 9 p2−1q4–2 = 9 pq2

(viii) (4x2y3)0 /  [ (x3)2 / x6 ] = 1 / [ x6 / x6 ] = 1 / 1 = 1

2. Division of an algebraic expression (polynomial) by a monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Example 3.7

Divide : (5y3 25y2 + 8 y) by 5y

Solution:

We have, (5y 3 25y2 + 8 y ) ÷ 5= 5y 3 25y2 + 8 y / 5y = y 3 15y2 1 + 8/5 = y 25y + 8/5

Think

Are the following divisions correct? Solution:

(i) [4y + 3] / 4 = y + 3 [4y + 3] / 4 = [4y / 4] + [3 / 4] = y + [3/4] is the correct answer.

The given answer is not correct.

(ii) [5m2 + 9] / 9 = 5m2 [5m2 + 9] / 9 = [5m2 / 9] + [9 / 9] = [ 5/9 m2 ] + 1 is the correct answer

The given answer is not correct.

(iii) [2x2 + 8] / 4 = 2x2 + 2 If not, correct it. [2x2 + 8] / 4 = [(2x2) / 4 ] + [ 8 / 4 ] = (1/2) x2 + 2 is the correct answer

The given answer is not correct.

Try these

(i) (16 y5 8y2 ) ÷ 4 y

(ii) ( p5 q2 + 24 p3 q 128q3 ) ÷ 6q

(iii) (4m2n + 9n2m + 3mn) ÷ 4mn

Solution: (i) (16y5 − 8y2) ÷ 4y

(16y5 − 8y2) / 4y = [16y5 / 4y ][ 8y2 / 4y ] = 4y5 – 1 − 2y2 – 1 = 4y4 – 2y

(ii) ( p5q2 + 24p3q − 128 q3 ) ÷ 6q

( p5q2 + 24p3q − 128 q3 ) / 6q  = [ p5q2 / 6q ] + [ 24p3q / 6q ][ 128q3 / 6q ]

= (1/6) p5q2 – 1 + 4p3q1 – 1 – (64/3)q3 – 1

= (1/6)p5q1 + 4p3q0 – (64/3)q2 = (1/6)p5q + 4p3 – (64/3)q2

(iii) ( 4m2n + 9n2m + 3mn ) ÷ 4mn

[ 4m2n + 9n2 m + 3mn ] / 4mn = [ 4m2n / 4mn ] + [ 9n2m / 4mn ] + [ 3mn / 4mn ]

= m2−1n1−1 + (9/4) m1−1n2−1 + (3/4)m1−1n1−1

= m1n0 + (9/4)m0 n1 + (3/4)m0n0 = m + (9/4)n + 3/4.    [ n0 =1]

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