An algebra problem by Calvin Lin

Algebra Level 1

Which of the following is the correct partial fraction decomposition of

x + 1 ( x + 2 ) ( x + 3 ) ? \frac{ x+1 } { ( x + 2) ( x + 3) } ?

2 x + 3 − 1 x + 2 \frac{2}{x+3} - \frac{ 1}{ x+2} 2 x + 3 + 1 x + 2 \frac{2}{ x+3} + \frac{ 1}{ x+2} 2 x + 3 − 1 x − 2 \frac{ 2}{ x+3} - \frac{ 1 } { x-2} 2 x − 3 − 1 x − 2 \frac{2}{x-3} - \frac{1}{ x-2}

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Calvin Lin by Calvin Lin

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1 solution

Joel Tan
Oct 21, 2014

Suppose it was written as a x + 2 + b x + 3 \frac {a}{x+2}+\frac {b}{x+3} where a , b a, b are real constants.

This is equal to a ( x + 3 ) + b ( x + 2 ) ( x + 2 ) ( x + 3 ) \frac {a (x+3)+b (x+2)}{(x+2)(x+3)} .

Comparing, a ( x + 3 ) + b ( x + 2 ) = a x + b x + 3 a + 2 b = ( a + b ) x + ( 3 a + 2 b ) = x + 1 a (x+3)+b (x+2)=ax+bx+3a+2b=(a+b)x+(3a+2b)=x+1 . Hence a + b = 1 ⟹ 3 a + 3 b = 3 , 3 a + 2 b = 1 a+b=1 \implies 3a+3b=3, 3a+2b=1

Solving by subtracting, b = ( 3 a + 3 b ) − ( 3 a + 2 b ) = 3 − 1 = 2 , a = 1 − 2 = − 1 b=(3a+3b)-(3a+2b)=3-1=2, a=1-2=-1 and we have the answer.

This method is efficient only if the choices were not given. Otherwise it is best to just fit in some values of x x and see which choice gives the correct answer.

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