Express (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions.
To express the given rational function (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions, we will follow these steps:
Step 1: Factorize the denominator.
The denominator (2-x)(1+2x^2) is already factored.
Step 2: Write the partial fraction form.
We need to determine the partial fraction form as (A/(2-x)) + (B(1+2x^2)).
Step 3: Find the common denominator.
The common denominator for A/(2-x) and B(1+2x^2) is (2-x)(1+2x^2).
Step 4: Expand the expression using the common denominator.
We can rewrite the original expression as [A(1+2x^2) + B(2-x)] / [(2-x)(1+2x^2)].
Step 5: Setting up the equation.
Since the numerator and denominator are polynomial expressions, they must be equal for all values of x. This gives us the equation:
15x^2 + 5x + 2 = A(1+2x^2) + B(2-x).
Step 6: Solve for A and B.
To solve for A and B, we will look at the coefficients of the corresponding terms on both sides of the equation.
For the constant term:
2 = B.
For the coefficient of x:
5 = -B + 0.
Setting these equations equal to each other, we get:
2 = -B.
Solving for B, we find B = -2.
For the coefficient of x^2:
15 = A + 2B.
Substituting the value of B, we get:
15 = A + 2(-2),
15 = A - 4.
Solving for A, we find A = 19.
Step 7: Write the partial fraction decomposition.
Now that we have found the values of A and B, we can rewrite the original expression as:
(15x^2 + 5x + 2) / [(2-x)(1+2x^2)] = (19/(2-x)) + (-2(1+2x^2))/(2-x)(1+2x^2).
Therefore, the partial fraction decomposition of (15x^2+5x+2)/(2-x)(1+2x^2) is: (19/(2-x)) + (-2(1+2x^2))/(2-x)(1+2x^2).