Question

Write the function as a sum of partial fractions. Do not determine the numerical values of the coefficients.(a) x/x^2+xâ2(b) 6âx/x^2â3xâ54

Answer 1

The correct equations in the question are:

(a) $\dfrac{x}{x^2+x-2}$   (b) $\dfrac{6-x}{x^2-3x-54}$

Answer:

Step-by-step explanation:

From above:

a) $\dfrac{x}{x^2+x-2}$  

We are to find the roots of the quadratic equation at the denominator.

i.e.

x²+x-2 =  x² + 2x - x - 2

= x(x+2) - 1(x+2)

= (x -1 ) (x + 2)

Thus;

$\mathbf{\dfrac{x}{x^2+x-2} = \dfrac{A}{x-1} + \dfrac{B}{x+2}}$

b) $\dfrac{6-x}{x^2-3x-54}$

The quadratic equation roots can be calculated by finding two numbers that if we multiply then, we will get -54 and if we add or subtract them, we will get -3

x²-3x -54 =  x² - 9x +6x - 54

=  x(x-9) + 6(x-9)

= (x + 6) (x - 9)

$\mathbf{\dfrac{6-x}{x^2-3x-54} = \dfrac{A}{x+6} + \dfrac{B}{x-9}}$