# Division of Algebraic Fractions

To solve the problems on division of algebraic fractions we will follow the same rules that we already learnt in dividing fractions in arithmetic.

From division of fractions we know,

First fraction ÷ Second fraction = First fraction × $$\frac{1}{Second fraction}$$

In algebraic fractions, the quotient can be determined in the same way i.e.

First algebraic fraction ÷ Second algebraic fraction

= First algebraic fraction × $$\frac{1}{Second algebraic fraction}$$

1. Determine the quotient of the algebraic fractions: $$\frac{p^{2}r^{2}}{q^{2}s^{2}} \div \frac{qr}{ps}$$

Solution:

$$\frac{p^{2}r^{2}}{q^{2}s^{2}} \div \frac{qr}{ps}$$

= $$\frac{p^{2}r^{2}}{q^{2}s^{2}} \times \frac{ps}{qr}$$

= $$\frac{p^{2}r^{2} \cdot ps}{q^{2}s^{2} \cdot qr}$$

= $$\frac{p^{3}r^{2}s}{q^{3}rs^{2}}$$

In the numerator and denominator of the quotient, the common factor is ‘rs’ by which if the numerator and denominator are divided, its lowest form will be = $$\frac{p^{3}r}{q^{3}s}$$

2. Find the quotient of the algebraic fractions: $$\frac{x(y + z)}{y^{2} - z^{2}} \div \frac{y + z}{y - z}$$

Solution:

$$\frac{x(y + z)}{y^{2} - z^{2}} \div \frac{y + z}{y - z}$$

= $$\frac{x(y + z)}{y^{2} - z^{2}} \times \frac{y - z}{y + z}$$

= $$\frac{x(y + z)}{(y + z)(y - z)} \times \frac{y - z}{y + z}$$

= $$\frac{x(y + z) \cdot (y - z)}{(y + z)(y - z) \cdot (y + z)}$$

= $$\frac{x(y + z)(y - z)}{(y + z)(y - z)(y + z)}$$

We observe that the common factor in the numerator and denominator of the quotient is (y + z) (y – z) by which, if the numerator and the denominator are divided, its lowest form will be $$\frac{x}{y + z}$$.

3. Divide the algebraic fractions and express in the lowest form:

$$\frac{m^{2} - m - 6}{m^{2} + 4m - 5} \div \frac{m^{2} - 4m + 3}{m^{2} + 6m + 5}$$

Solution:

$$\frac{m^{2} - m - 6}{m^{2} + 4m - 5} \div \frac{m^{2} - 4m + 3}{m^{2} + 6m + 5}$$

= $$\frac{m^{2} - m - 6}{m^{2} + 4m - 5} \times \frac{m^{2} + 6m + 5}{m^{2} - 4m + 3}$$

= $$\frac{m^{2} - 3m + 2m - 6}{m^{2} + 5m - m - 5} \times \frac{m^{2} + 5m + m + 5}{m^{2} - 3m - m + 3}$$

= $$\frac{m(m - 3) + 2(m - 3)}{m(m + 5) - 1(m + 5)} \times \frac{m(m + 5) + 1(m + 5)}{m(m - 3) - 1(m - 3)}$$

= $$\frac{(m - 3)(m + 2)}{(m + 5) (m - 1)} \times \frac{(m + 5) (m + 1)}{(m - 3) (m - 1)}$$

= $$\frac{(m - 3)(m + 2) \cdot (m + 5) (m + 1)}{(m + 5) (m - 1) \cdot (m - 3) (m - 1)}$$

= $$\frac{(m - 3)(m + 2)(m + 5) (m + 1)}{(m + 5) (m - 1)(m - 3) (m - 1)}$$

We observe that the common factor in the numerator and denominator of the quotient is (m - 3) (m + 5), by which if the numerator and the denominator of the quotient is divided, $$\frac{(m + 2) (m + 1)}{(m - 1) (m - 1)}$$ i.e. $$\frac{(m + 2) (m + 1)}{(m - 1)^{2}}$$ will be its reduced lowest form.

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