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 *x ^{m}* ÷

__1. Division
of a monomial by another monomial__

Dividing
a monomial 10 *p*^{4} by another
monomial 2*p*^{3}, we get

10p^{4} ÷ 2p^{3}

However,
to divide we can also follow laws of exponents as,

**Think**

Are the following correct?

(i) *x*^{3}/*x*^{8} = *x*^{8-3} = *x*^{5}

(ii) 10*m*^{4} /
10*m*^{4 }= 0

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

**Solution:**

**(i) x^{3} / x^{8} = x^{8
– 3} = x^{5}**

*x*^{3} / *x*^{8}
= *x*^{3 – 8} = *x ^{−}*

∴ The given answer is
wrong.

**(ii) 10 m^{4} / 10m^{4} = 0**

10*m*^{4} / 10*m*^{4} = [10/10] *m*^{4−4}
= 1 *m*^{0} = 1 [ ∵ *m*^{0} = 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* = *x*^{1 – 1} = *x*^{0}
= 1

∴ The given statement is correct

**Example 3.6**

Velu pastes
‘4*xy* ’ pictures in one page of his scrap
book. How many pages will he need to paste 100*x*^{2} *y*^{3}
pictures? (*x, y* are positive integers)

*Solution:*

Total number
of pictures = 100*x*^{2} *y*^{3}

Pictures
in one page = 4*xy*

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

= 25*xy*^{2} pages

**Try these**

**Divide**

(i) 12*x* ^{3}*y*^{2} by *x*^{2}*y *(ii) −20*a*^{5}*b*^{2} by 2*a*^{3}*b*^{7} (iii) 28*a*^{4} *c*^{2} by 21*ca*^{2}

(iv) (3 *x* ^{2}
*y*)^{3} √6*x*^{2}*y*^{3 }(v) 64*m*^{4} (*n*^{2})^{3}
÷ 4*m*^{2}*n*^{2 }(vi) (8 *x* ^{2} *y* ^{2} )^{3} ÷ (8*x* ^{2} *y*^{2} )^{2}

(vii) 81 *p*^{2}*q*^{4} ÷ √[81*p*^{2}*q*^{4 }]^{}

(vii) ( 4*x*^{2}* y*^{3}* *)^{0}
÷ [( *x*^{3})^{2}]/*x*^{6}

**Solution:**^{}

^{}

**Solution:**

**(i) **12*x*^{3}*y*^{2}
/ *x*^{2}*y* = 12*x*^{3−2 }*y*^{2−1} = 12*x*^{1}*y*^{1}
= 12 *xy*

**(ii) **−20*a*^{5}*b*^{2}
/ 2*a*^{3}*b*^{7} = −20*a*^{5 − 3} / 2*b*^{7
− 2 } = −10*a*^{2} / *b*^{5}

**(iii) **28* a*^{4}*c*^{2}
/ 21*ca*^{2} = [28 / 21] *a*^{4 – 2} *c*^{2 –
1} = 4/3 *a*^{2}*c*^{1} = 4/3 *a*^{2}*c*

**(iv) **(3*x*^{2}*y*)^{3}
/ 6*x*^{2}*y*^{3} = 27 *x*^{6}*y*^{3}
/ 6 *x*^{2}*y*^{3 } = [9/2] *x*^{6–2} = 9/2 *x*^{4}

**(v) **64*m*^{4}(*n*^{2})^{3}
/ 4*m*^{2}*n*^{2}** = **64*m*^{4}*n*^{6}
/ 4*m*^{2}*n*^{2} = 16 *m*^{4−2} *n*^{6−2}
= 16 *m*^{2}*n*^{4}

**(vi) **(8*x*^{2}*y*^{2})^{3}
/ (8*x*^{2}*y*^{2})^{2} = 512 *x*^{6}*y*^{6}
/ 64* x*^{4}*y*^{4}

= 8*x*^{6−4}* y*^{6−4}

**= 8***x*^{2}*y*^{2}

**(vii) **81*p*^{2}*q*^{4}
/ √81*p*^{2}*q*^{4} = 81*p*^{2}*q*^{4}
/ 9*pq*^{2} = 9 *p*^{2−1}*q*^{4–2} = 9 *pq*^{2}

**(viii)** (4*x*^{2}*y*^{3})^{0}
/ [ (*x*^{3})^{2} /
*x*^{6} ] = 1 / [ *x*^{6} / *x*^{6} ] =
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 :
(5*y*^{3} −
25*y*^{2} +
8 *y*) by 5*y*

*Solution:*

We have,
(5*y* ^{3} −
25*y*^{2} +
8 *y* ) ÷ 5*y *_{=} 5*y* ^{3} − 25*y*^{2} + 8 *y / *5*y*

= *y* ^{3} ^{−}^{1} − 5*y*^{2} ^{−}^{1} + 8/5 = *y* ^{2} − 5*y* + 8/5

**Think**

Are the following divisions correct?

**Solution:**

**(i) [4 y + 3] / 4 = y + 3**

** [**4*y* + 3] / 4 = [4*y */ 4] + [3 / 4] = *y* + [3/4]
is the correct answer.

∴ The given answer is not
correct.

**(ii) [5 m^{2} + 9] / 9 = 5m^{2}**

^{}

[5*m*^{2} + 9] / 9 = [5*m*^{2} / 9] + [9
/ 9] = [ 5/9 *m*^{2} ] + 1 is the correct answer

∴ The given answer is not
correct.

**(iii) [2 x^{2} + 8] / 4 = 2x^{2} +
2 If not, correct it.**

[2*x*^{2} + 8] / 4 = [(2*x*^{2}) / 4 ]
+ [ 8 / 4 ] = (1/2) *x*^{2} + 2 is the correct answer

∴ The given answer is not
correct.

**Try these**

(i) (16 *y*^{5} − 8*y*^{2}
) ÷ 4 *y*

(ii) ( *p*^{5} *q*^{2} + 24 *p*^{3}
*q* −128*q*^{3} ) ÷ 6*q*

(iii) (4*m*^{2}*n* + 9*n*^{2}*m* + 3*mn*) ÷ 4*mn*

**Solution:**

**(i) ****(16 y^{5 }− 8y^{2}) ÷ 4y**

(16*y*^{5 }− 8*y*^{2}) / 4y = [16*y*^{5}
/ 4*y *]*− *[ 8*y*^{2
}/ 4y ] = 4*y*^{5 – 1 }− 2*y*^{2 – 1} = 4*y*^{4}
– 2*y*

**(ii) ( ***p*^{5}*q*^{2 }**+ 24 p^{3}q − 128 q^{3} ) ÷
6q**

( *p*^{5}*q*^{2 }+ 24*p*^{3}*q* −
128* q*^{3} ) / 6*q* =
[ *p*^{5}*q*^{2 }/ 6*q *] + [ 24*p*^{3}*q* / 6*q *]* – *[ 128*q*^{3}
/ 6*q *]

= (1/6) *p*^{5}*q*^{2 – 1 }+ 4*p*^{3}*q*^{1
– 1} – (64/3)*q*^{3 – 1}

= (1/6)*p*^{5}*q*^{1 }+ 4*p*^{3}*q*^{0}
– (64/3)*q*^{2} = (1/6)*p*^{5}*q*^{ }+ 4*p*^{3}
– (64/3)*q*^{2}

**(iii) ( ****4 m^{2}n + 9n^{2}m +
3mn ) ÷ 4mn**

[ 4*m*^{2}*n* + 9*n*^{2} *m*
+ 3*mn* ] / 4*mn* = [ 4*m*^{2}*n* / 4*mn* ] + [
9*n*^{2}*m* / 4*mn* ] + [ 3*mn */ 4*mn *]

= *m*^{2−1}*n*^{1−1} + (9/4)* m*^{1−1}*n*^{2−1}
+ (3/4)*m*^{1−1}*n*^{1−1}

= *m*^{1}*n*^{0} + (9/4)*m*^{0}
*n*^{1} + (3/4)*m*^{0}*n*^{0 }*=* *m*
+ (9/4)*n* + 3/4. [ ∵ *n*^{0 }*=*1]

Tags : Algebra | Chapter 3 | 8th Maths , 8th Maths : Chapter 3 : Algebra

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