Answer: To express each of the given rational expressions in partial fractions, we need to factor the denominators and then use the method of partial fractions decomposition. Let's go through each one:
a. (2x³ - x² + x + 5) / (x² + 3x + 2)
Step 1: Factor the denominator
x² + 3x + 2 = (x + 2)(x + 1)
Step 2: Express the given rational expression in partial fractions:
(2x³ - x² + x + 5) / (x² + 3x + 2) = A/(x + 2) + B/(x + 1)
Step 3: Find the values of A and B:
2x³ - x² + x + 5 = A(x + 1) + B(x + 2)
Now, equate the coefficients of x³, x², x, and the constant term on both sides:
Coefficient of x³: 0 = A
Coefficient of x²: 2 = -A + B
Coefficient of x: 1 = A + 2B
Constant term: 5 = B
Solving the system of equations, we get:
A = 0
B = 5
So, the partial fraction decomposition is:
(2x³ - x² + x + 5) / (x² + 3x + 2) = 0/(x + 2) + 5/(x + 1)
b. (x² - x) / (x² + x + 1)
Step 1: Factor the denominator
x² + x + 1 does not factor further.
Step 2: Express the given rational expression in partial fractions:
(x² - x) / (x² + x + 1) = A/(x² + x + 1) + Bx/(x² + x + 1)
Step 3: Find the values of A and B:
x² - x = A + Bx
Now, equate the coefficients of x² and x on both sides:
Coefficient of x²: 1 = A
Coefficient of x: -1 = B
So, the partial fraction decomposition is:
(x² - x) / (x² + x + 1) = 1/(x² + x + 1) - x/(x² + x + 1)
c. (x + 6) / (x³ + 3x² - 4x + 12)
Step 1: Factor the denominator
x³ + 3x² - 4x + 12 does not factor further.
Step 2: Express the given rational expression in partial fractions:
(x + 6) / (x³ + 3x² - 4x + 12) = A/(x³ + 3x² - 4x + 12)
Step 3: Find the value of A:
x + 6 = A(x³ + 3x² - 4x + 12)
Now, equate the coefficients of x³, x², x, and the constant term on both sides:
Coefficient of x³: 0 = A
Coefficient of x²: 1 = 3A
Coefficient of x: 1 = -4A
Constant term: 6 = 12A
Solving the system of equations, we get:
A = 1/2
So, the partial fraction decomposition is:
(x + 6) / (x³ + 3x² - 4x + 12) = 1/(2x³ + 3x² - 4x + 12)
d. (-x + 7) / ((x + 2)(x - 1)²)
Step 1: Factor the denominator
(x + 2)(x - 1)² does not factor further.
Step 2: Express the given rational expression in partial fractions:
(-x + 7) / ((x + 2)(x - 1)²) = A/(x + 2) + B/(x - 1) + C/(x - 1)²
Step 3: Find the values of A, B, and C:
-x + 7 = A(x - 1)² + B(x + 2)(x - 1) + C(x + 2)
Now, equate the coefficients of x², x, and the constant term on both sides:
Coefficient of x²: 0 = A + B
Coefficient of x: -1 = -2A + C
Constant term: 7 = A + 2B + 2C
Solving the system of equations, we get:
A = 1
B = -1
C = 0
So, the partial fraction decomposition is:
(-x + 7) / ((x + 2)(x - 1)²) = 1/(x + 2) - 1/(x - 1)
These are the partial fractions for each given rational expression.