Express in partial fraction.7x^2-18x-7 divided by (x-4)(2x^2-6x+3)
To express the given rational expression, 7x^2 - 18x - 7, in partial fractions, we first need to factorize the denominator, (x - 4)(2x^2 - 6x + 3).
1. Factorize the quadratic term: To factorize 2x^2 - 6x + 3, we can use the quadratic formula or complete the square. In this case, the quadratic cannot be factorized further, so we leave it as it is.
2x^2 - 6x + 3 = (2x + 1)(x - 1)
2. Set up the partial fraction expression: Since the denominator has two distinct irreducible factors, we can express the rational expression as:
7x^2 - 18x - 7 / [(x - 4)(2x^2 - 6x + 3)] = A / (x - 4) + (Bx + C) / (2x^2 - 6x + 3)
Here, A, B, and C are constants that we need to determine.
3. Clear the denominator: Multiply both sides by the denominator to clear it:
7x^2 - 18x - 7 = A(2x^2 - 6x + 3) + (Bx + C)(x - 4)
Expand both sides and combine like terms:
7x^2 - 18x - 7 = 2Ax^2 - 6Ax + 3A + Bx^2 - 5Bx - 4Cx + 4B + C
4. Match the coefficients: Since the equation should hold for all values of x, we can equate the coefficients of corresponding powers of x on both sides.
Coefficients of x^2:
7 = 2A + B
Coefficients of x:
-18 = -6A - 5B - 4C
Coefficients of the constant term:
-7 = 3A + 4B + C
5. Solve the system of equations: Solve the system of three equations to find the values of A, B, and C.
From the first equation, 7 = 2A + B, we can solve for B:
B = 7 - 2A
Substitute B = 7 - 2A into the second equation:
-18 = -6A - 5(7 - 2A) - 4C
-18 = -6A -35 + 10A - 4C
-18 = 4A - 4C - 35
17 = 4A - 4C
From the third equation, -7 = 3A + 4B + C, we can substitute B = 7 - 2A:
-7 = 3A + 4(7 - 2A) + C
-7 = 3A + 28 - 8A + C
-35 = -5A + C
We have a system of three equations with three unknowns: A, B, and C.
6. Solve the system of equations: Solve the system of equations formed by the equations 17 = 4A - 4C, and -35 = -5A + C to obtain the values of A and C.
Multiply the second equation by 4 and add it to the first equation:
68 - 20A + 4C + 17 = 0
85 - 20A + 4C = 0
Substitute the value of C from Equation 2 into the equation above:
85 - 20A + 4(-35 + 5A) = 0
85 - 20A - 140 + 20A = 0
-55 = 0
Since -55 ≠ 0, there is no solution for this system of equations.
7. Thus, we cannot express the given rational expression in partial fractions since the system of equations does not have a solution.
Please note that the given denominator may be incorrect or mistyped.