Express (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions.
To express the fraction (15x^2+5x+2)/((2-x)(1+2x^2)) in partial fractions, we need to break it down into simpler fractions with denominators of 2-x and 1+2x^2.
Step 1: Factorize the denominator
The given denominator (2-x)(1+2x^2) cannot be simplified further, so we move on to the next step.
Step 2: Assume the Partial Fractions form
We assume that the given fraction can be expressed as:
(15x^2+5x+2)/((2-x)(1+2x^2)) = A/(2-x) + (Bx+C)/(1+2x^2)
Step 3: Cross-multiply and solve for unknowns
Now, we need to find the values of A, B, and C. We can achieve this by equating the numerators of both sides of the equation.
15x^2 + 5x + 2 = A(1+2x^2) + (Bx+C)(2-x)
Multiplying out on both sides gives:
15x^2 + 5x + 2 = A + 2Ax^2 + 2Bx - Bx^2 + 2Cx - Cx^2
Rearranging the equation in descending order of powers of x, we have:
(2A - B)x^2 + (2B + 2C)x + (A + 2C) = 15x^2 + 5x + 2
By comparing coefficients of like terms, we get the following equations:
2A - B = 15 (equation 1)
2B + 2C = 5 (equation 2)
A + 2C = 2 (equation 3)
Solving these three equations will give us the values of A, B, and C.
Step 4: Solve the system of equations
We can solve this system of equations using various methods such as substitution, elimination, or matrices. Here, we will use the substitution method.
Solve equation 3 for A:
A = 2 - 2C
Substitute the value of A in equation 1:
2(2 - 2C) - B = 15
4 - 4C - B = 15
-B - 4C = 11 (equation 4)
Substitute the value of A in equation 2:
2B + 2C = 5
2(2 - 2C) + 2C = 5
4 - 4C + 2C = 5
4 - 2C = 5
-2C = 1
C = -1/2
Substitute the value of C in equation 4:
-B - 4(-1/2) = 11
-B + 2 = 11
-B = 9
B = -9
Substitute the values of B and C in equation 3 to find A:
A + 2C = 2
A + 2(-1/2) = 2
A - 1 = 2
A = 3
So, A = 3, B = -9, and C = -1/2.
Step 5: Express the original fraction in partial fractions
Now that we have found the values of A, B, and C, we can express the original fraction in partial fractions:
(15x^2+5x+2)/((2-x)(1+2x^2)) = 3/(2-x) + (-9x -1/2)/(1+2x^2)
Therefore, the expression (15x^2+5x+2)/((2-x)(1+2x^2)) can be written as 3/(2-x) + (-9x -1/2)/(1+2x^2) in partial fractions.