**Dividing Rational Expressions :**

Thus division of one rational expression by other is equivalent to the product of first and reciprocal of the second expression. If the resulting expression is not in its lowest form then reduce to its lowest form.

**Step 1 :**

Factor both numerator and denominator, if possible.

**Step 2 :**

Identify the common factor at both numerator and denominator.

**Step 3 : **

The common factor identified at both numerator and denominator should be multiplied by the other terms.

**Step 4 :**

Now, get rid of the common factor at both numerator and denominator.

**Question 1 :**

(i) Simplify

[(2a^{2}+5a+3)/(2a^{2}+7a+6)] ÷ [(a^{2}+6a+5)/(-5a^{2}-35a-50)]

**Solution :**

2a^{2 }+ 5a + 3 = (a + 1)(2a + 3)

2a^{2 }+ 7a + 6 = (2a + 3)(a + 2)

a^{2 }+ 6a + 5 = (a + 1)(a + 5)

-5a^{2}-35a-50 = -5(a^{2} + 7a + 10)

= -5(a + 2)(a + 5)

= [(a + 1)(2a + 3)/(2a + 3)(a + 2)] ÷ [(a + 1)(a + 5)/-5(a + 2)(a + 5)]

= [(a + 1)(2a + 3)/(2a + 3)(a + 2)] ⋅ [-5(a + 2)(a + 5)/(a + 1)(a + 5)]

= -5

(ii) [(b^{2} + 3b - 28)/(b^{2} + 4b + 4)] ÷ [(b^{2} - 49)/(b^{2} - 5b - 14)]

**Solution :**

= (b^{2} + 3b - 28)/(b^{2} + 4b + 4)] ÷ [(b^{2} - 49)/(b^{2} - 5b - 14)

b^{2} + 3b - 28 = (b - 4)(b + 7)

b^{2} + 4b + 4 = (b + 2)(b + 2)

b^{2} - 49 = b^{2} - 7^{2} = (b - 7)(b + 7)

b^{2} - 5b - 14 = (b - 7) (b + 2)

= [(b-4)(b+7)/(b+2)(b+2)] ÷ [(b-7)(b+7)/(b-7) (b+2)]

= [(b-4)(b+7)/(b+2)(b+2)] ⋅ [(b-7) (b+2)/(b-7)(b+7)]

= (b - 4)/(b + 2)

(iii) [(x + 2)/4y] ÷ [(x^{2} - x - 6)/12y^{2}]

**Solution :**

(x^{2} - x - 6) = (x - 3) (x + 2)

= [(x + 2)/4y] ÷ [(x - 3) (x + 2)/12y^{2}]

= [(x + 2)/4y] ⋅ [12y^{2}/(x - 3) (x + 2)]

= 3y/4(x - 3)

(iv) [(12t^{2} - 22t + 8)/3t] ÷ [(3t^{2} + 2t - 8)/(2t^{2}+4t)]

**Solution :**

12t^{2} - 22t + 8 = 2(6t^{2} - 11t + 4)

= 2(3t - 4)(2t - 1)

(3t^{2} + 2t - 8) = (t + 2)(3t - 4)

2t^{2}+4t = 2t(t + 2)

= [(12t^{2} - 22t + 8)/3t] ÷ [(3t^{2} + 2t - 8)/(2t^{2}+4t)]

= [2(3t - 4)(2t - 1)/3t] ÷ [(t + 2)(3t - 4)/2t(t + 2)]

= [2(3t - 4)(2t - 1)/3t] ⋅ [2t(t + 2)/(t + 2)(3t - 4)]

= 4(2t - 1)/3

**Question 2 :**

If x = (a^{2} + 3a - 4)/(3a^{2} - 3) and y = (a^{2} + 2a - 8)/(2a^{2} - 2a - 4) find the value of x^{2}y^{-2}

Solution :

x = (a^{2} + 3a - 4)/(3a^{2} - 3)

= (a + 4)(a - 1)/3(a^{2} - 1)

= (a + 4)(a - 1)/3(a + 1)(a - 1)

x^{2} = [(a + 4)/3(a + 1)]^{2}

y = (a^{2} + 2a - 8)/(2a^{2} - 2a - 4)

= (a + 4)(a - 2)/2(a - 2)(a + 1)

= (a + 4)/2(a + 1)

y^{-}^{2} = [(a + 4)/2(a + 1)]^{-}^{2}

x^{2}y^{-2 = }x^{2}/y^{2}

= [(a + 4)/3(a + 1)]^{2}/ [(a + 4)/2(a + 1)]^{2}

= (a + 4)^{2}/9(a + 1)^{2}/ [(a + 4)^{2}/4(a + 1)^{2}]

= (a + 4)^{2}/9(a + 1)^{2}⋅ [4(a + 1)^{2}/(a + 4)^{2}]

= 4/9

**Question 3 :**

If a polynomial p(x) = x^{2} −5x −14 is divided by another polynomial q(x) we get (x - 7)/(x + 2), find q (x).

**Solution :**

p(x)/q(x) = (x - 7)/(x + 2)

(x^{2} −5x −14) / q(x) = (x - 7)/(x + 2)

(x - 7)(x + 2) / q(x) = (x - 7)/(x + 2)

q(x) = (x - 7)(x + 2) /(x - 7)/(x + 2)

q(x) = 1

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