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Solved Examples

Question 1: Find GCD of 125 - 5x2 and x2 + 3x - 10
Solution:
Step 1:
Factorized polynomials

125 - 5x2 = 5(25 - x2 )

= 5(52 - x2 )

= 5(5 - x)(5 + x)       [a2 - b2 = (a - b)(a + b)]

and
x2 + 3x - 10 = x2 + 5x - 2x  - 10

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

= (x - 2)(x + 5)

Step 2:
125 - 5x2  = 5(5 - x)(5 + x)

x2 + 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

x3 - 27 = x3 - 33

= (x - 3)(x2 + 3x + 9)

[a3 - b3 = (a - b)(a2 + ab + b2 )]

and
5x2 - 13x - 6 = 5x2 - 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

=> y = 18, put in equation (3)

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

=> x = 12

Hence the numbers are 12 and 18.