Express (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions.
To express the given rational expression in partial fractions, follow these steps:
Step 1: Factorize the denominator.
The denominator (2-x)(1+2x^2) can be expressed as (2-x)(1+sqrt(2)x)(1-sqrt(2)x).
Step 2: Write down the given expression as the sum of partial fractions.
(15x^2+5x+2)/(2-x)(1+2x^2) = A/(2-x) + B/(1+sqrt(2)x) + C/(1-sqrt(2)x). Here, A, B, and C are constants that we need to find.
Step 3: Multiply both sides of the equation by the denominator.
(15x^2+5x+2) = A(1+sqrt(2)x)(1-sqrt(2)x) + B(2-x)(1-sqrt(2)x) + C(2-x)(1+sqrt(2)x).
Step 4: Simplify and group like terms.
To find the values of A, B, and C, we need to compare the coefficients of the corresponding powers of x on both sides of the equation.
On the left-hand side (LHS), the coefficient of x^2 is 15, the coefficient of x is 5, and the constant term is 2.
On the right-hand side (RHS):
- The term with (1+sqrt(2)x)(1-sqrt(2)x) has no x terms, so the coefficient of x^2 is 0.
- The term with (2-x)(1-sqrt(2)x) has a coefficient of x, which is -(A + B).
- The term with (2-x)(1+sqrt(2)x) has a coefficient of x, which is -(A + C).
- The constant term is 2A + 2B + 2C.
Step 5: Set up and solve a system of equations.
By equating the coefficients on the LHS and RHS, we get the following equations:
0 = 0 (coefficients of x^2)
5 = -(A + B) (coefficients of x)
2 = 2A + 2B + 2C (constant term)
Solving these equations will yield the values of the constants A, B, and C.
Step 6: Substitute the values of A, B, and C back into the equation:
(15x^2+5x+2)/(2-x)(1+2x^2) = A/(2-x) + B/(1+sqrt(2)x) + C/(1-sqrt(2)x).
That's the expression in partial fractions for the given rational expression.