Question

The partial fraction decomposition of 28/ x^2−4 can be written in the form of f(x)/ x−2 + g(x)/ x+2, where f(x)? where g(x)?

Answer 1

The partial fraction decomposition of 28/ (x^2-4) can be written as:

f(x) = -7 / (x-2) + 7 / (x+2)

Step-by-step explanation:

Factor the denominator: First, factor the denominator of the given fraction: 28 / (x^2 - 4) = 28 / (x-2)(x+2).

Assume the form: Since the denominator is factored into two linear terms, we assume the partial fraction decomposition to be:

28 / (x-2)(x+2) = f(x) / (x-2) + g(x) / (x+2)

where f(x) and g(x) are polynomials of a degree one less than the degree of the polynomial in the denominator (in this case, both f(x) and g(x) can be constants).

Clear denominators: To eliminate the fractions, we multiply both sides of the equation by the common denominator (x-2)(x+2):

28 = f(x)(x+2) + g(x)(x-2)

Solve for f(x) and g(x): Substitute values of x that will eliminate one of the unknown variables and solve for the other.

Substitute x = 2: 28 = f(2)(2+2) + g(2)(2-2) => 28 = 4f(2) => f(2) = 7.

Substitute x = -2: 28 = f(-2)(-2+2) + g(-2)(-2-2) => 28 = -4g(-2) => g(-2) = -7.

Substitute back and simplify: Substitute the values of f(2) and g(-2) back into the equation to solve for f(x) and g(x):

f(x) = 7 / (x-2)

g(x) = -7 / (x+2)

Therefore, the complete partial fraction decomposition of 28/ (x^2-4) is:

28 / (x^2-4) = -7 / (x-2) + 7 / (x+2)

Answer 2

The value of f(x) is - 7 , and The value of g(x) is 7 .

Step-by-step explanation:

Given as :

The partial fraction fraction decomposition of
`(28)/(x^(2) - 4)`

i.e
`(28)/(x^(2) - 4)` =
`(f(x))/(x-2)` +
`(g(x))/(x+2)`

Or,
`(28)/(x^(2) - 4)` =
`(f(x)* (x-2) + g(x)* (x+2))/(x^(2)-4)`

Or, 28 = f(x) × (x - 2) + g(x) × (x + 2)

Now, put x = 2

So , 28 = f(x) × (2 - 2) + g(x) × (2 + 2)

Or, 28 = 0 + 4 × g(x)

∴ g(x) =
`(28)/(4)` = 7

Now, put x = - 2

So , 28 = f(x) × ( - 2 - 2) + g(x) × ( - 2 + 2)

Or, 28 = - 4 × f(x) + g(x) × 0

∴ f(x) =
`(28)/( - 4)` = - 7

Hence The value of f(x) is - 7 , and The value of g(x) is 7 . Answer