REDUCING RATIONAL EXPRESSIONS TO LOWEST TERMS

Reducing Rational Expressions to Lowest Terms :

A rational expression p (x) /q (x) is said to be in its lowest form if GCD(p(x),q(x)) = 1.

To reduce a rational expression to its lowest form, follow the given steps

(i) Factorize the numerator and the denominator

(ii) If there are common factors in the numerator and denominator, cancel them.

(iii) The resulting expression will be a rational expression in its lowest form.

Reducing Rational Expressions to Lowest Terms - Questions

Question 1 :

Reduce each of the following rational expressions to its lowest form.

(i)  (x2 - 1) / (x2 + x)

Solution :

  =  (x2 - 1) / (x2 + x)  

x2 - 1  =  (x + 1) (x - 1)

(x2 + x)  = x(x + 1)

  =  (x + 1) (x - 1) / x(x + 1)

  =  (x - 1)/x

(ii)  (x2 - 11x + 18)/(x2 - 4x + 4)

Solution :

  =   (x2 - 11x + 18)/(x2 - 4x + 4)

(x2 - 11x + 18)  =  (x - 9)(x - 2)

(x2 - 4x + 4)  =  (x - 2)(x - 2)

  =   (x - 9)(x - 2)/(x - 2)(x - 2)

  =  (x - 9)/(x - 2)

(iii)  (9x2 + 81x) / (x3 + 8x2 - 9x)

Solution :

  =   (9x2 + 81x) / (x3 + 8x2 - 9x)

(9x2 + 81x)  =  9x(x + 9)

(x3 + 8x2 - 9x)  =  x(x2 + 8x - 9)

  =  x(x + 9)(x - 1)

  =   9x(x + 9) / x(x + 9)(x - 1)

  =  9/(x - 1)

(iv)  (p2 - 3p - 40)/(2p3-24p2+64p)

Solution :

  =  (p2 - 3p - 40)/(2p- 24p+ 64p)

p2 - 3p - 40  =  (p - 8)(p + 5)

2p- 24p+ 64p  =  2p(p2 - 12p + 32)

  =  2p (p -8) (p - 4)

  =  (p - 8)(p + 5)/2p (p -8) (p - 4)

  =  (p + 5)/2p (p - 4)

After having gone through the stuff given above, we hope that the students would have understood, "Reducing Rational Expressions to Lowest Terms". 

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