x-1/x^2-x-20
Give the equivalent numerator if the denominator is (x - 5)(x + 4)(x + 3).
I do not understand what you are asking.
What is an equivalent numerator?
The numerator, x^2-x-20 , is actually (x-5)(x+4)
To find the equivalent numerator, we can use partial fraction decomposition. The given denominator is (x - 5)(x + 4)(x + 3). Let's begin by factoring the numerator, x - 1.
We start by factorizing the denominator:
(x - 5)(x + 4)(x + 3)
Now, we need to express the given rational expression as a sum of partial fractions with unknown numerators, with each denominator corresponding to one of the factors in the factorized denominator.
So, we assume that the equivalent numerator can be expressed as:
A/(x - 5) + B/(x + 4) + C/(x + 3)
To find the unknowns A, B, and C, we can proceed by multiplying the entire equation by the common denominator (x - 5)(x + 4)(x + 3):
(x - 1) = A(x + 4)(x + 3) + B(x - 5)(x + 3) + C(x - 5)(x + 4)
Expanding and simplifying the equation, we get:
x - 1 = A(x^2 + 7x + 12) + B(x^2 - 2x - 15) + C(x^2 - x - 20)
Now, let's equate the coefficients of the same powers of x on both sides of the equation.
For x^2: 0 = A + B + C
For x: -1 = 7A - 2B - C
For constant terms: -1 = 12A - 15B - 20C
We now have a system of three equations with three unknowns. Solving the system, we find the values of A, B, and C.
Solving the first equation for A, we get A = -B - C
Substituting A = -B - C into the second equation, we have:
-1 = 7(-B - C) - 2B - C
-1 = -7B - 7C - 2B - C
-1 = -9B - 8C
Rearranging terms, we get:
9B + 8C = 1 --------> Equation 1
Similarly, substituting A = -B - C into the third equation, we have:
-1 = 12(-B - C) - 15B - 20C
-1 = -12B - 12C - 15B - 20C
-1 = -27B - 32C
Rearranging terms, we get:
27B + 32C = -1 --------> Equation 2
Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (B and C).
Solving the system of equations, we find the values of B and C.
Once we have the values of B and C, we can substitute them back into A = -B - C to find the value of A.
Finally, we can express the given rational expression as the sum of partial fractions:
(x - 1)/(x^2 - x - 20) = A/(x - 5) + B/(x + 4) + C/(x + 3)
Now the equivalent numerator has been obtained, which is A for the factor (x - 5), B for the factor (x + 4), and C for the factor (x + 3).