Express in partial fraction.7x^2-18x-7 divided by (x-4)(2x^2-6x+3)
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To express the rational function 7x^2 - 18x - 7 divided by (x-4)(2x^2 - 6x + 3) in partial fractions, you first need to factor the denominator completely.
The denominator (x-4)(2x^2 - 6x + 3) is already factored.
The next step is to determine the degree of the numerator and denominator. The degree of the numerator is 2 (quadratic) and the degree of the denominator is 3 (cubic).
Since the degree of the numerator is less than the degree of the denominator, we can set up the partial fraction decomposition as follows:
7x^2 - 18x - 7 = A/(x-4) + (Bx + C)/(2x^2 - 6x + 3)
To find the constants A, B, and C, we need to perform the process of equating coefficients.
First, we multiply through by the denominator:
7x^2 - 18x - 7 = A(2x^2 - 6x + 3) + (Bx + C)(x - 4)
Expanding and collecting like terms, we get:
7x^2 - 18x - 7 = 2Ax^2 - 6Ax + 3A + Bx^2 - 4Bx + Cx - 4C
Now, let's group the terms with the same degree:
(7x^2 - 18x - 7) = (2Ax^2 + Bx^2) + (-6Ax - 4Bx + Cx) + (3A - 4C)
Comparing the coefficients of like terms on both sides of the equation, we obtain the following system of equations:
2A + B = 7 (coefficient of x^2 terms)
-6A - 4B + C = -18 (coefficient of x terms)
3A - 4C = -7 (constant term)
Now, we can solve this system of equations to find the values of A, B, and C.
Solving these equations, we find:
A = 1
B = 5
C = -6
Therefore, the partial fraction decomposition of 7x^2 - 18x - 7 / ((x-4)(2x^2 - 6x + 3)) is:
(1/(x-4)) + ((5x - 6)/(2x^2 - 6x + 3))