The following steps will be useful to add and subtract rational expressions.
Step 1 :
Check whether the denominators of two rational expressions are same.
Step 2 :
After taking least common multiple the denominator will become same. So, put only one denominator and combine the numerators.
Example :
Simplify :
[(x + 2)/(x + 3)] + [(x - 1)/(x + 3)]
Solution :
Because the denominators are same, we take the denominator once and combine the numerators.
Example :
Simplify :
[(x + 1)/(x - 1)2] + [1/(x + 1)]
Solution :
To make the denominators same, we need to take least common multiple.
Question 1 :
Simplify :
[x3 / (x - 2)] + [8 / (2-x)]
Answer :
= [x3 / (x - 2)] + [8 / (2-x)]
= [x3 / (x - 2)] - [8 / (x-2)]
= [ (x³ - 8)/(x - 2) ]
By comparing x3 - 8 with the algebraic identity
a3 - b3 = (a - b) (a2 + ab + b2)
we get,
= (x - 2) (x2 + x(2) + 22) / (x - 2)
= (x2 + 2 x + 4)
Question 2 :
Simplify :
(x + 2)/(x2 + 3x + 2)] + (x - 3)/(x2 - 2x - 3)
Solution :
= [(x + 2)/(x2 + 3x + 2)] + [(x - 3)/(x2 - 2x - 3)]
(x² + 3 x + 2) = (x + 1) (x + 2)
(x² - 2 x - 3) = (x - 3) (x + 1)
Question 3 :
Simplify
[(x2 - x - 6)/(x2 - 9)] + [(x2 + 2x - 24)/(x2 - x - 12)]
Solution :
= [(x2 - x - 6)/(x2 - 9)] + [(x2 + 2x - 24)/(x2 - x - 12)]
(x2 - x - 6) = (x - 3) (x + 2)
By comparing (x2 - 9) with the algebraic identity
(a2 - b2) = (a + b) (a - b)
we get,
(x2 - 32) = (x + 3)(x - 3)
(x2 + 2 x - 24) = (x + 6) (x - 4)
(x2 - x - 12) = (x - 4) (x + 3)
Question 4 :
Simplify
[(x - 2)/(x2 - 7x + 10)] + [(x + 3)/(x2 - 2x - 15)]
Solution :
= [(x - 2)/(x2 - 7x + 10)] + [(x + 3)/(x2 - 2x - 15)]
(x² - 7 x + 10) = (x - 2 ) (x - 5)
(x² - 2 x - 15) = (x - 5) (x + 3)
Question 5 :
Simplify
[(2x2-5x+3)/(x2-3x+2)] - [(2x2-7x-4)/(2x2 - 3x - 2)]
Solution :
= [(2x2-5x+3)/(x2-3x+2)] - [(2x2-7x-4)/(2x2 - 3x - 2)]
Question 6 :
Simplify
[(x2-4)/(x2+6x+8)]-[(x2-11x+30)/(x2-x - 20)]
Solution :
= [(x2-4)/(x2+6x+8)] - [(x2-11x+30)/(x2-x - 20)]
(x2 - 22) = (x + 2) (x - 2)
(x2+ 6x + 8) = (x + 2) (x + 4)
(x2- 11x + 30) = (x - 6) (x - 5)
(x2- x - 20) = (x - 5) (x + 4)
Question 7 :
Simplify
[(2x + 5)/(x + 1)] + [(x2 + 1)/(x2 - 1)] - [(3x - 2)/(x - 1)]
Solution :
Question 8 :
Simplify
[1/(x2+3x+2)] + [1/(x2+5x+6)] - [2/(x2+4x+3)]
Solution :
(x2 + 3x + 2) = (x + 1) (x + 2)
(x2 + 5x + 6) = (x + 2)(x + 3)
(x2 + 4x + 3) = (x + 3) (x + 1)
= 0
Question 9 :
Which rational expression should be added to
(x3 - 1)/(x2 + 2) to get (3x3 + 2x2 + 4)/(x2 + 2) ?
Solution :
let the required rational expression be p(x)
[(x3 - 1)/(x2 + 2)] + p(x) = (3x3 + 2x2 + 4)/(x2 + 2)
p(x) = [(3x3 + 2x2 + 4)/(x2 + 2)] - [(x3 - 1)/(x2 + 2)]
Since the denominators are same, we may write only one denominator and combine the numerators.
= [(3x3 + 2x2 + 4) - (x3 - 1)]/(x2 + 2)]
= (3x3 - x3 + 2x2 + 4 + 1)/(x2 + 2)
= (2x3 + 2x2 + 5)/(x2 + 2)
Question 10 :
Which rational expression should be subtracted from
(4x3 - 7 x2 + 5)/(2x - 1) to get 2x2 - 5x + 1 ?
Solution :
Let p(x) be the required rational expression
(4x3 - 7 x2 + 5)/(2x - 1) - p(x) = 2x2 - 5x + 1
(4x3 - 7 x2 + 5)/(2x - 1) - (2x2 - 5x + 1) = p(x)
p(x) = [(4x3 - 7 x2 + 5) - (2x2 - 5x + 1)(2x - 1)]/(2x - 1)
Question 11 :
If P = x/(x + y) Q = y/(x + y), then find
[1/(P-Q)] - [2Q/(P²-Q²)]
Solution :
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