Express (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions.
To express a rational function in partial fractions, we first need to factor the denominators. In this case, we have (2-x) and (1+2x^2).
1. The first step is to factor the denominators completely:
- (2-x) can be rewritten as -(x-2)
- (1+2x^2) cannot be simplified further
2. Now, let's express the given rational function in partial fractions:
Let's assume it can be expressed as:
A/(2-x) + B/(1+2x^2)
3. Next, adding the fractions back together, we have:
(A(1+2x^2) + B(2-x))/((2-x)(1+2x^2))
4. Equating the numerators on both sides of the equation, we have:
15x^2 + 5x + 2 = A(1 + 2x^2) + B(2 - x)
5. Expanding and combining like terms, we get:
15x^2 + 5x + 2 = A + 2Ax^2 + 2B - Bx
6. By comparing the coefficients of the corresponding terms on both sides,
we get a system of equations:
2A - B = 5 (coefficient of x term)
2A + B = 2 (coefficient of x^2 term)
-A + 2B = 2 (constant term)
7. Solving this system of equations, we find:
A = 1, B = 3
8. Finally, substituting the values of A and B back into the partial fraction expression:
(15x^2 + 5x + 2)/((2 - x)(1 + 2x^2)) = 1/(2 - x) + 3/(1 + 2x^2)
So, the given rational function can be expressed in partial fractions as:
(15x^2 + 5x + 2)/((2 - x)(1 + 2x^2)) = 1/(2 - x) + 3/(1 + 2x^2)