How do I write the partial fraction decomposition of 7/(x^2+13x+40)?
To write the partial fraction decomposition of 7/(x^2+13x+40), you can follow these steps:
Step 1: Factorize the denominator:
First, factorize the quadratic expression x^2+13x+40. This can be done by finding two numbers that multiply together to give 40 and add up to 13. In this case, the factors are (x+5) and (x+8):
x^2+13x+40 = (x+5)(x+8)
Step 2: Write the partial fraction decomposition:
Now, you can express the fraction 7/(x^2+13x+40) as the sum of two fractions with separate denominators:
7/(x^2+13x+40) = A/(x+5) + B/(x+8)
Step 3: Finding the values of A and B:
To find the values of A and B, you need to make the fractions on the right side of the equation have a common denominator. Multiply A/(x+5) by (x+8)/(x+8) and B/(x+8) by (x+5)/(x+5):
7/(x^2+13x+40) = (A(x+8))/(x+5)(x+8) + (B(x+5))/(x+5)(x+8)
Now, you can equate the numerators on both sides of the equation to find the values of A and B:
7 = A(x+8) + B(x+5)
Next, simplify the equation by expanding the brackets:
7 = Ax + 8A + Bx + 5B
Combine like terms:
7 = (A+B)x + (8A+5B)
Since the equation should hold true for all values of x, equate the coefficients of x and the constants on both sides of the equation:
A + B = 0 (for the x term)
8A + 5B = 7 (for the constant term)
Now, you have a system of two equations with two unknowns. Solve this system of equations to find the values of A and B.
the fractions will be
A/(x+8) + B/(x+5)
A(x+5) + B(x+8) = 7
(A+B)x + (5A+8B) = 0x + 7
A+B = 0
5A+8B = 7
3B = 7
B = 7/3
A = -7/3
7/(x^2+13x+40) = (7/3)/(x+5) - (7/3)/(x+8))