write the form of the partial decomposition for the given rational expression. Don't solve for constants:
8x+10/x^2+7x-18
please show work!
I am pretty sure you meant:
(8x+10)/(x^2 + 7x - 18)
= (8x+10)/((x+9)(x-2))
let This be
A/(x+9) + B(x-2)
= ( A(x-2) + B(x+9) )/( (x+9)(x-2) )
so ...
A(x-2) + B(x+9) = 8x + 10
sub in A = 2, and B=-9 to find the constanst
To find the partial fraction decomposition of the given rational expression, we need to factor the denominator and then determine the form of the decomposition based on the factors. Let's start by factoring the denominator:
x^2 + 7x - 18
To factor this quadratic expression, we need to find two numbers whose sum is 7 and whose product is -18. After a bit of trial and error, we can factor it as:
(x + 9)(x - 2)
Now that we have factored the denominator, we can write the partial fraction decomposition in the following form:
8x + 10 / (x + 9)(x - 2) = A / (x + 9) + B / (x - 2)
Here, A and B are the constants that we need to determine. Notice that for each factor in the denominator, we have a corresponding term in the numerator.
To find the values of A and B, we can multiply both sides of the equation by the common denominator:
8x + 10 = A(x - 2) + B(x + 9)
Next, we can simplify and collect like terms:
8x + 10 = (A + B)x + (9B - 2A)
Now, we can compare the coefficients of x on both sides of the equation:
8 = A + B (1)
0 = 9B - 2A (2)
We can now solve this system of equations to find the values of A and B. By solving equations (1) and (2), we can find the values of A and B without solving for constants.
Once we have the values of A and B, we can substitute them back into the partial fraction decomposition:
8x + 10 / (x + 9)(x - 2) = A / (x + 9) + B / (x - 2)
Remember, we have determined the form of the partial fraction decomposition but we haven't actually solved for the constants at this point.