Express (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions.
To express the given expression as partial fractions, we need to factorize the denominator and then find the unknown constants that will make up the numerator.
Step 1: Factorize the denominator:
(2 - x)(1 + 2x^2)
Step 2: Write the expression as a sum of two fractions:
(15x^2 + 5x + 2) / ((2 - x)(1 + 2x^2)) = A / (2 - x) + (Bx + C) / (1 + 2x^2)
Step 3: Find the values of A, B, and C:
To find the values of A, B, and C, we need to clear the denominators.
Multiplying both sides of the equation by the common denominator (2 - x)(1 + 2x^2), we get:
15x^2 + 5x + 2 = A(1 + 2x^2) + (Bx + C)(2 - x)
Now, we need to expand and group like terms:
15x^2 + 5x + 2 = A + 2Ax^2 + 2Bx - Bx^2 + 2Cx - Cx^2
Rearranging the terms:
15x^2 + 5x + 2 = (2A - B)x^2 + (2B + 2C)x + (A - C)
Matching the coefficients of each power of x on both sides of the equation, we can create a system of equations:
2A - B = 15 (coefficients of x^2)
2B + 2C = 5 (coefficients of x)
A - C = 2 (constant terms)
Solving this system of equations, we find:
A = 4/3
B = -7/6
C = 2/3
So, the partial fraction decomposition of the given expression is:
(15x^2 + 5x + 2) / ((2 - x)(1 + 2x^2)) = (4/3) / (2 - x) + (-7/6)x + (2/3) / (1 + 2x^2)