Question
use partial fraction decomposition to rewrite the function f(x)=(11x-2)/(6x^(2)+x-1)
Answers
GPT 3.5
To perform partial fraction decomposition on the function f(x)=(11x-2)/(6x^2+x-1), we first need to factor the denominator.
The denominator 6x^2+x-1 can be factored as (3x-1)(2x+1).
Now we can write the function f(x) as a sum of two fractions:
f(x) = A/(3x-1) + B/(2x+1)
Next, we need to find the values of A and B.
Multiplying both sides of the equation by the denominator (3x-1)(2x+1), we get:
(11x-2) = A(2x+1) + B(3x-1)
Expanding and simplifying the equation, we have:
11x - 2 = 2Ax + A + 3Bx - B
Matching the coefficients of the powers of x on both sides of the equation, we get:
11x = 2Ax + 3Bx
-2 = A - B
From the coefficients, we can determine that A = 3 and B = -5.
Therefore, the partial fraction decomposition of f(x) is:
f(x) = 3/(3x-1) - 5/(2x+1)
The denominator 6x^2+x-1 can be factored as (3x-1)(2x+1).
Now we can write the function f(x) as a sum of two fractions:
f(x) = A/(3x-1) + B/(2x+1)
Next, we need to find the values of A and B.
Multiplying both sides of the equation by the denominator (3x-1)(2x+1), we get:
(11x-2) = A(2x+1) + B(3x-1)
Expanding and simplifying the equation, we have:
11x - 2 = 2Ax + A + 3Bx - B
Matching the coefficients of the powers of x on both sides of the equation, we get:
11x = 2Ax + 3Bx
-2 = A - B
From the coefficients, we can determine that A = 3 and B = -5.
Therefore, the partial fraction decomposition of f(x) is:
f(x) = 3/(3x-1) - 5/(2x+1)