decompose 5x+1/x^2-x-12 into partial functions?
You posted this question twice. See my other answer. You should have parentheses around the numerator and denominator. Otherwise, it can be confused with
5x + 1/x^2 -x -12, which is NOT the same.
To decompose the rational function 5x + 1 / (x^2 - x - 12) into partial fractions, you need to factor the denominator and express the rational function as a sum of simpler fractions with unknown coefficients. Here's how you can do it step by step:
Step 1: Factor the denominator (x^2 - x - 12) into two linear factors:
The quadratic equation x^2 - x - 12 can be factored as (x - 4)(x + 3).
Step 2: Express the rational function as a sum of partial fractions:
5x + 1 / (x^2 - x - 12) can be expressed as A / (x - 4) + B / (x + 3), where A and B are the unknown coefficients.
Step 3: Find the values of A and B by finding a common denominator and equating the numerators:
Multiply both sides of the equation by (x - 4)(x + 3):
5x + 1 = A(x + 3) + B(x - 4)
Distribute and simplify:
5x + 1 = (A + B)x + 3A - 4B
Equate the coefficients of like terms on both sides of the equation:
5x = (A + B)x (coefficient of x)
1 = 3A - 4B (constant term)
Step 4: Solve the system of equations to find the values of A and B:
From the first equation, since both sides must be equal, A + B = 5.
From the second equation, 3A - 4B = 1.
Solve these two equations simultaneously to find the values of A and B:
A + B = 5 (Equation 1)
3A - 4B = 1 (Equation 2)
There are several ways to solve this system of equations, such as substitution or elimination. Let's use substitution.
Solving equation 1 for A, we get A = 5 - B.
Substitute the value of A in equation 2:
3(5 - B) - 4B = 1
15 - 3B - 4B = 1
-7B = -14
B = 2
Substitute the value of B in equation 1 to find A:
A + 2 = 5
A = 3
Step 5: Write the partial fraction decomposition:
The decomposition of 5x + 1 / (x^2 - x - 12) into partial fractions is:
5x + 1 / (x^2 - x - 12) = 3 / (x - 4) + 2 / (x + 3)
That's it! You have successfully decomposed the given rational function into partial fractions.