x-1/x^2-x-20 Give the equivalent numerator if the denominator is (x - 5)(x + 4)(x + 3).
I bet you mean
(x-1)/(x^2-x-20)
= (x-1)/( (x-5)(x+4) ) , but you want the denominator to be (x - 5)(x + 4)(x + 3)
So how does it differ? , so make the changes by multiplying top and bottom by (x+3)
To find the equivalent numerator, we need to decompose the given rational expression into partial fractions.
First, let's factorize the denominator (x - 5)(x + 4)(x + 3):
(x - 5)(x + 4)(x + 3) = (x)(x)(x) + ((-5)(x)(x) + (4)(x)(x) + (3)(x)(x)) + (-5)(4)(x) + (-5)(3)(x) + (4)(3)(x) + (-5)(4)(3)
= x^3 + (-5x^2 + 4x^2 + 3x^2) + (-20x - 15x + 12x) + (-60)
= x^3 + 2x^2 - 23x - 60
Now, let's express the given rational expression, x - 1/x^2 - x - 20, in terms of partial fractions:
x - 1/x^2 - x - 20 = A/(x - 5) + B/(x + 4) + C/(x + 3)
Multiplying both sides by the common denominator, (x - 5)(x + 4)(x + 3), we get:
(x - 5)(x + 4)(x + 3)(x - 1/x^2 - x - 20) = A(x + 4)(x + 3) + B(x - 5)(x + 3) + C(x - 5)(x + 4)
Expanding and simplifying the equation:
(x^3 + 2x^2 - 23x - 60)(x) - (x^2 - x - 20)(x - 5)(x + 4)(x + 3) = A(x + 4)(x + 3) + B(x - 5)(x + 3) + C(x - 5)(x + 4)
x^4 + 2x^3 - 23x^2 - 60x - x^4 + 5x^3 + 4x^2 - 20x - x^3 + 5x^2 + 4x - 20 = A(x^2 + 7x + 12) + B(x^2 - 2x - 15) + C(x^2 - x - 20)
Combining like terms:
-15x^2 - 76x - 20 = A(x^2 + 7x + 12) + B(x^2 - 2x - 15) + C(x^2 - x - 20)
Now, we can equate coefficients to solve for A, B, and C.
For the x^2 terms:
A + B + C = 0
For the x terms:
7A - 2B - C = -76
For the constant terms:
12A - 15B - 20C = -20
Solving the system of equations, we find:
A = 4, B = -1, C = -3
Therefore, the equivalent numerator is:
4/(x - 5) - 1/(x + 4) - 3/(x + 3)