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.

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

Question 1 :

(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

Question 2 :

(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)

Question 3 :

(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)

Question 4 :

(p² - 3p - 40) / (2p³-24p²+64p)

Solution :

= (p² - 3p - 40)/(2p³- 24p²+ 64p)

p² - 3p - 40 = (p - 8)(p + 5)

2p³- 24p²+ 64p = 2p(p² - 12p + 32)

= 2p (p -8) (p - 4)

= (p - 8)(p + 5)/2p (p -8) (p - 4)

= (p + 5)/2p (p - 4)

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