Question

partial fraction : (1+2x)(1+3x)(1+4x)/(1-2x)(1-3x)(1-4x)

Answer 1

(1+2x)(1+3x) (1+4x)
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
(1-2x) (1-3x) (1-4x)
=1
Because each one will be canceled by each other
Eg: (1+2x)
¯¯¯¯¯¯
(1-2x)

= 1
Therefore the answer is 1

Answer 2

Answer:

$\frac{30}{(1-2x)} - \frac{70}{(1-3x)} + \frac{42}{(1-4x)}$

Step-by-step explanation:

Given:- (1+2x)(1+3x)(1+4x)/(1-2x)(1-3x)(1-4x)

To Find:- Partial Fraction of the given expression.

Solution:-

Partial Fraction is obtained when a rational expression is split into sum of two or more rational expression. Every factor present in the denominator of a rational expression corresponds to a partial fraction. The denominator of the expression needs to be factorized to obtain the set of partial fractions.

Steps that need to be followed to do so are:

1. First we factorize the numerator and denominator to simplify the expression.

2. Then we split the rational expression based on the numerator and denominator expression.

3. Then we take the LCM of the factors of the denominators and multiply both sides of the equation with this LCM.

4. Simplify and get the values of A and B by comparing coefficients of like terms on both sides.

5. Then we substitute the values of A and B on the right side of the equation to obtain the partial fraction.

The formula is P/((ax + b)2 = [A/(ax + b)] + [B/(ax + b)2]

⇒  $\frac{(1+2x)(1+3x)(1+4x)}{(1-2x)(1-3x)(1-4x)} = \frac{A}{(1-2x)} + \frac{B}{(1-3x)} + \frac{C}{(1-4x)}$

⇒ $\frac{(1+2x)(1+3x)(1+4x)}{(1-2x)(1-3x)(1-4x)} = \frac{A(1-3x)(1-4x)+ B(1-2x)(1-4x)+C(1-2x)(1-3x)}{(1-2x)(1-3x)(1-4x)}$

⇒ (1+2x)(1+3x)(1+4x) = A(1-3x)(1-4x) + B(1-2x)(1-4x) + C(1-2x)(1-3x)  ------ (1)

Putting x = 1/2 in above equation, we get

⇒ 2(1+3/2)(1+2)  = A(1-3/2)(1-2) + 0 + 0

⇒ 6 × 5/2 = A( -1/2 )( -1 )

⇒ A = 30.

Putting x = 1/3 in equation ( 1 ), we get  

⇒ (1 + 2/3)(1 + 1)(1 + 4/3) = 0 + B(1 - 2/3)(1 - 4/3) + 0

⇒ (5/3)( 2 )(7/3) = B( 1/3 )( -1/3 )

⇒ B = ( 70/9 ) × ( -9)

⇒ B = -70.

Putting x = 1/4 in equation ( 1 ), we get

⇒ (1 + 1/2)(1 + 3/4)(1 + 1) = 0 + 0 + C(1 - 1/2)(1 - 3/4)

⇒ (3/2)(7/4)2 = C(1/2)(1/4)

⇒ C = ( 42/8 ) × ( 8 )

⇒ C = 42.

∴ The partial fraction becomes = $\frac{30}{(1-2x)} - \frac{70}{(1-3x)} + \frac{42}{(1-4x)}$

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