Write out the partial fraction decomposition of the following rational expression.
1.) 3x^2-7x-2/x^3-x
2.) x^3-x+3/x^2+x-2
I'll do one, and you can do the other. You can check your work at wolframalpha.com
x^3-x = x(x-1)(x+1) so we want
A/x + B/(x-1) + C/(x+1) = (3x^2-7x-2)/(x^3-x)
So, using the common denominator, we have
A(x^2-1) + Bx(x+1) + Cx(x-1) = 3x^2-7x-2
(A+B+C)x^2 + (B-C)x - A = 3x^2-7x-2
If they are identical, then
A+B+C = 3
B-C = -7
-A = -2
So,
A = 2
B = -3
C = 4
and we have
2/x - 3/(x-1) + 4/(x+1)
check at
http://www.wolframalpha.com/input/?i=partial+fractions+%283x^2-7x-2%29%2F%28x^3-x%29
Sure! I can help you with that. Let's start with the first expression:
1.) To find the partial fraction decomposition of (3x^2 - 7x - 2) / (x^3 - x), we follow these steps:
Step 1: Factorize the denominator (x^3 - x):
x(x^2 - 1)
x(x - 1)(x + 1)
Step 2: Write the expression as the sum of partial fractions:
(3x^2 - 7x - 2) / (x(x - 1)(x + 1))
A/x + B/(x - 1) + C/(x + 1)
Step 3: Clear the denominators by multiplying both sides of the equation by (x)(x - 1)(x + 1):
(3x^2 - 7x - 2) = A(x - 1)(x + 1) + Bx(x + 1) + Cx(x - 1)
Step 4: Expand and simplify:
3x^2 - 7x - 2 = A(x^2 - 1) + B(x^2 + x) + C(x^2 - x)
Step 5: Collect like terms:
3x^2 - 7x - 2 = (A + B + C)x^2 + (B - C)x - A
Now, we can equate the coefficients of like terms on both sides of the equation:
Coefficient of x^2: A + B + C = 3
Coefficient of x: B - C = -7
Coefficient of constant term: -A = -2
From the third equation, we get A = 2. Substituting this value into the second equation, we get B - C = -7. We can rearrange this equation to C = B + 7.
Finally, substituting the value of A and C back into the first equation, we get 2 + B + B + 7 = 3. Simplifying, we find 2B + 9 = 3.
Solving for B, we get B = -3.
Since we have the values of A, B, and C, we can write the partial fraction decomposition as:
(3x^2 - 7x - 2) / (x^3 - x) = 2/x + (-3)/(x - 1) + (-3)/(x + 1)
Now, let's move on to the second expression:
2.) To find the partial fraction decomposition of (x^3 - x + 3) / (x^2 + x - 2), we follow these steps:
Step 1: Factorize the denominator (x^2 + x - 2):
(x + 2)(x - 1)
Step 2: Write the expression as the sum of partial fractions:
(x^3 - x + 3) / (x + 2)(x - 1)
A/(x + 2) + B/(x - 1)
Step 3: Clear the denominators by multiplying both sides of the equation by (x + 2)(x - 1):
x^3 - x + 3 = A(x - 1) + B(x + 2)
Step 4: Expand and simplify:
x^3 - x + 3 = Ax - A + Bx + 2B
Step 5: Collect like terms:
x^3 - x + 3 = (A + B)x + (-A + 2B + 3)
Now, we can equate the coefficients of like terms on both sides of the equation:
Coefficient of x: A + B = -1 (coefficient of x on the left side is -1)
Coefficient of constant term: -A + 2B + 3 = 3 (constant term on the left side is 3)
From the second equation, we get -A + 2B + 3 = 3. Simplifying, we find -A + 2B = 0.
Since A = 2B, we can substitute this value into the first equation: A + B = -1. Substituting A = 2B, we have 2B + B = -1. Solving for B, we find B = -1/3.
Now, substituting the value of B into A = 2B, we get A = 2(-1/3). Simplifying, we find A = -2/3.
Since we have the values of A and B, we can write the partial fraction decomposition as:
(x^3 - x + 3) / (x^2 + x - 2) = (-2/3)/(x + 2) + (-1/3)/(x - 1)
To find the partial fraction decomposition of a rational expression, follow these steps:
1. Factor the denominator of the rational expression as much as possible.
2. Write down the partial fraction decomposition using unknown constants as coefficients for each factor in the denominator.
3. Determine the values of the unknown constants by equating the original rational expression with the sum of the partial fractions.
Let's work through the examples:
1.) 3x^2 - 7x - 2 / (x^3 - x)
Step 1: Factor the denominator:
x^3 - x = x(x^2 - 1) = x(x + 1)(x - 1)
Step 2: Write the partial fraction decomposition:
3x^2 - 7x - 2 / (x^3 - x) = A / x + B / (x + 1) + C / (x - 1)
Step 3: Determine the values of the unknown constants:
Multiplying through by the denominator, we have:
3x^2 - 7x - 2 = A(x + 1)(x - 1) + Bx(x - 1) + Cx(x + 1)
Expanding and collecting like terms:
3x^2 - 7x - 2 = A(x^2 - 1) + B(x^2 - x) + C(x^2 + x)
Matching coefficients of x^2, x, and the constant term, we get the following system of equations:
A + B + C = 3 (coefficient of x^2)
-A - B + C = -7 (coefficient of x)
-A + C = -2 (constant term)
Solving these equations, we find A = 1, B = 2, and C = 0.
Therefore, the partial fraction decomposition of 3x^2 - 7x - 2 / (x^3 - x) is:
(1 / x) + (2 / (x + 1))
2.) x^3 - x + 3 / (x^2 + x - 2)
Step 1: Factor the denominator:
x^2 + x - 2 = (x + 2)(x - 1)
Step 2: Write the partial fraction decomposition:
x^3 - x + 3 / (x^2 + x - 2) = A / (x + 2) + B / (x - 1)
Step 3: Determine the values of the unknown constants:
Multiplying through by the denominator, we have:
x^3 - x + 3 = A(x - 1) + B(x + 2)
Expanding and collecting like terms:
x^3 - x + 3 = Ax - A + Bx + 2B
Matching coefficients of x^3, x, and the constant term, we get the following system of equations:
A + B = 1 (coefficient of x^3)
-A + 2B = -1 (coefficient of x)
-A + 2B = 3 (constant term)
Solving these equations, we find A = 1/3 and B = 2/3.
Therefore, the partial fraction decomposition of x^3 - x + 3 / (x^2 + x - 2) is:
(1/3) / (x + 2) + (2/3) / (x - 1)