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Question

If $fraction numerator x to the power of 4 end exponent over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals P left parenthesis x right parenthesis plus fraction numerator A over denominator x minus a end fraction plus fraction numerator B over denominator x minus b end fraction plus fraction numerator C over denominator x minus c end fraction$ then $P left parenthesis x right parenthesis equals$

$x - a$
$x - a - b$
$x - a - b - c$
$x + a + b + c$

hint Hint:

In order to integrate a rational function, it is reduced to a proper rational function. The method in which the integrand is expressed as the sum of simpler rational functions is known as decomposition into partial fractions. After splitting the integrand into partial fractions, it is integrated accordingly with the help of traditional integrating techniques.

The correct answer is: $x plus a plus b plus c$

Given :
$fraction numerator x to the power of 4 end exponent over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals P left parenthesis x right parenthesis plus fraction numerator A over denominator x minus a end fraction plus fraction numerator B over denominator x minus b end fraction plus fraction numerator C over denominator x minus c end fraction$
To Find : P(x)
Let P(x) be a linear polynomial i.e. = mx + n ( since y = mx + c)
Substitute this value in the equation
$fraction numerator x to the power of 4 over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals m x space plus space n plus fraction numerator A over denominator x minus a end fraction plus fraction numerator B over denominator x minus b end fraction plus fraction numerator C over denominator x minus c end fraction$

Taking LCM
$fraction numerator x to the power of 4 over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction equals fraction numerator left parenthesis m x space plus space n right parenthesis left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis space plus space A left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis plus space B left parenthesis x minus a right parenthesis left parenthesis x minus c right parenthesis space plus space C left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis space space over denominator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis end fraction$

Cancelling denominators on both sides
$x to the power of 4 over blank equals fraction numerator left parenthesis m x space plus space n right parenthesis left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis space plus space A left parenthesis x minus b right parenthesis left parenthesis x minus c right parenthesis plus space B left parenthesis x minus a right parenthesis left parenthesis x minus c right parenthesis space plus space C left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis space space over denominator blank end fraction$

On comparing the coefficients of $x to the power of 4 space comma space x cubed space comma x squared space a n d space x$ from both sides ,we get
$rightwards double arrow$ m = 1
$rightwards double arrow$ -am -bm -cm + n = 0
$rightwards double arrow$ a + b + c = n
$rightwards double arrow$ P(x) =mx + n
Substituting the values of m and n in
$rightwards double arrow$ P(x) = x + a + b + c

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