use partial fraction decomposition to rewrite the function f(x)=(11x-2)/(6x^(2)+x-1)
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)
To rewrite the function f(x) using partial fraction decomposition, we need to factor the denominator first.
The denominator of f(x) is a quadratic equation: 6x^2 + x - 1. We can factor it as follows:
6x^2 + x - 1 = (2x - 1)(3x + 1)
Now we can rewrite f(x) using the partial fraction decomposition by assuming that it can be expressed as the sum of two fractions with unknown denominators A and B:
f(x) = (11x - 2) / ((2x - 1)(3x + 1))
= A / (2x - 1) + B / (3x + 1)
To find the values of A and B, we need to simplify the right side and equate the numerators:
(11x - 2) = A(3x + 1) + B(2x - 1)
Expanding the right side and rearranging the equation:
11x - 2 = (3A + 2B)x + (A - B)
Now we can equate the coefficients of the corresponding powers of x:
11x - 2 = (3A + 2B)x + (A - B)
Equating the coefficients of x:
11 = 3A + 2B (1)
Equating the constant terms:
-2 = A - B (2)
Now we have a system of two equations (1) and (2) with two unknowns, A and B. To solve this system, we can use substitution or elimination.
From equation (2), we can rewrite it as:
A = 2 - B
Substitute this expression for A into equation (1):
11 = 3(2 - B) + 2B
Simplify:
11 = 6 - 3B + 2B
Combine like terms:
11 = 6 - B
Add B to both sides:
B + 11 = 6
Subtract 6 from both sides:
B = -5
Now substitute this value of B back into equation (2):
A = 2 - (-5) = 2 + 5 = 7
So, we have found the values of A = 7 and B = -5.
Now we can rewrite f(x) using partial fraction decomposition:
f(x) = 7 / (2x - 1) - 5 / (3x + 1)
Therefore, the function f(x) can be written as the sum of two separate fractions using partial fraction decomposition.
To rewrite the function `f(x) = (11x - 2)/(6x^2 + x - 1)` using partial fraction decomposition, follow these steps:
Step 1: Factorize the denominator of the given function.
The denominator `6x^2 + x - 1` can be factored as `(2x - 1)(3x + 1)`.
Step 2: Express the given function in terms of partial fractions.
The given function can be expressed as:
`f(x) = (A/(2x - 1)) + (B/(3x + 1))`
Step 3: Solve for the unknown constants A and B.
Multiply through by the common denominator `(2x - 1)(3x + 1)` to obtain:
`11x - 2 = A(3x + 1) + B(2x - 1)`
Expanding and collecting like terms:
`11x - 2 = (3A + 2B)x + (A - B)`
Equating the coefficients of x on both sides:
11 = 3A + 2B ...(1)
Equating the constant terms on both sides:
-2 = A - B ...(2)
Solve the system of equations (1) and (2) to find the values of A and B.
Step 4: Solve the system of equations.
From equation (2), we have A - B = -2.
Solving equation (2) for A, we get: A = B - 2.
Substitute the value of A in equation (1):
11 = 3A + 2B
11 = 3(B - 2) + 2B
11 = 3B - 6 + 2B
11 = 5B - 6
5B = 17
B = 17/5
Substituting the value of B in equation (2):
A = B - 2
A = (17/5) - 2
A = (17/5) - (10/5)
A = 7/5
Step 5: Write the partial fraction decomposition.
The function can be rewritten using partial fraction decomposition as:
`f(x) = (7/5)/(2x - 1) + (17/5)/(3x + 1)`
Therefore, the rewritten function using partial fraction decomposition is `f(x) = (7/5)/(2x - 1) + (17/5)/(3x + 1)`.