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### Solved Examples

**Question 1:**Find GCD of 125 - 5x

^{2}and x

^{2}+ 3x - 10

**Solution:**

Step 1:

Factorized polynomials

125 - 5x

= 5(5

= 5(5 - x)(5 + x) [a

and

x

= x(x + 5) - 2(x + 5)

= (x - 2)(x + 5)

Step 2:

125 - 5x

x

=> The required GCD = x + 5. answer

Factorized polynomials

125 - 5x

^{2}= 5(25 - x^{2})= 5(5

^{2}- x^{2})= 5(5 - x)(5 + x) [a

^{2}- b^{2}= (a - b)(a + b)]and

x

^{2}+ 3x - 10 = x^{2}+ 5x - 2x - 10= x(x + 5) - 2(x + 5)

= (x - 2)(x + 5)

Step 2:

125 - 5x

^{2}= 5(5 - x)(5 + x)x

^{2}+ 3x - 10 = (x - 2)(x + 5)=> The required GCD = x + 5. answer

**Question 2:**Express in lowest term

$\frac{x^3 - 27}{5x^2 - 13x - 6}$

**Solution:**

Step 1:

Factorized polynomials

x

= (x - 3)(x

[a

and

5x

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

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

Step 2:

$\frac{x^3 - 27}{5x^2 - 13x - 6}$ = $\frac{ (x - 3)(x^2 + 3x + 9)}{(5x + 2)(x - 3)}$

= $\frac{x^2 + 3x + 9}{5x + 2}$

=> $\frac{x^3 - 27}{5x^2 - 13x - 6}$ = $\frac{x^2 + 3x + 9}{5x + 2}$

Factorized polynomials

x

^{3}- 27 = x^{3}- 3^{3}= (x - 3)(x

^{2}+ 3x + 9)[a

^{3}- b^{3}= (a - b)(a^{2}+ ab + b^{2})]and

5x

^{2}- 13x - 6 = 5x^{2}- 15x + 2x - 6= 5x(x - 3) + 2(x - 3)

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

Step 2:

$\frac{x^3 - 27}{5x^2 - 13x - 6}$ = $\frac{ (x - 3)(x^2 + 3x + 9)}{(5x + 2)(x - 3)}$

= $\frac{x^2 + 3x + 9}{5x + 2}$

=> $\frac{x^3 - 27}{5x^2 - 13x - 6}$ = $\frac{x^2 + 3x + 9}{5x + 2}$

**Question 3:**Two numbers are in the ratio 2:3. If 5 is subtracted from the numerator and 3 added to the denominator, the ratio becomes 1:3. Find the numbers.

**Solution:**

Step 1:

Let the fraction is $\frac{x}{y}$

=> $\frac{x}{y}$ = $\frac{2}{3}$ ......................(1)

also given, when 5 is subtracted from the numerator and 3 added to the denominator, the ratio becomes 1:3.

=> $\frac{x - 5}{y + 3}$ = $\frac{1}{3}$ ........................(2)

Step 2:

(1) => x = $\frac{2}{3}$y ............................(3)

and (2) => 3(x - 5) = y + 3

=> 3x - 15 = y + 3

=> 3x - y = 18 ................................(4)

Step 3:

Put x = $\frac{2}{3}$y in equation (4)

=> 3 * $\frac{2}{3}$ y - y = 18

=> 2y - y = 18

=>

=> x = $\frac{2}{3}$ * 18

=>

Hence the numbers are

Let the fraction is $\frac{x}{y}$

=> $\frac{x}{y}$ = $\frac{2}{3}$ ......................(1)

also given, when 5 is subtracted from the numerator and 3 added to the denominator, the ratio becomes 1:3.

=> $\frac{x - 5}{y + 3}$ = $\frac{1}{3}$ ........................(2)

Step 2:

(1) => x = $\frac{2}{3}$y ............................(3)

and (2) => 3(x - 5) = y + 3

=> 3x - 15 = y + 3

=> 3x - y = 18 ................................(4)

Step 3:

Put x = $\frac{2}{3}$y in equation (4)

=> 3 * $\frac{2}{3}$ y - y = 18

=> 2y - y = 18

=>

**y = 18,**put in equation (3)=> x = $\frac{2}{3}$ * 18

=>

**x = 12**Hence the numbers are

**12 and 18.**