Questions: Give the partial fraction decomposition for the following function. 20x-7/x^2-x 20x-7/x^2-x=

$Give the partial fraction decomposition for the following function. 20x-7/x^2-x 20x-7/x^2-x=$

Transcript text: Give the partial fraction decomposition for the following function. \[ \begin{array}{l} \frac{20 x-7}{x^{2}-x} \\ \frac{20 x-7}{x^{2}-x}= \end{array} \]

Solution

Solution Steps

To find the partial fraction decomposition, first factor the denominator. Then, express the fraction as a sum of simpler fractions with unknown coefficients. Finally, solve for these coefficients by equating and simplifying.

Step 1: Factor the Denominator

The given function is $\frac{20x - 7}{x^2 - x}$. First, factor the denominator: \[ x^2 - x = x(x - 1) \]

Step 2: Set Up Partial Fractions

Express the fraction as a sum of simpler fractions: \[ \frac{20x - 7}{x(x - 1)} = \frac{A}{x} + \frac{B}{x - 1} \]

Step 3: Solve for Coefficients

Multiply through by the common denominator $x(x - 1)$ to clear the fractions: \[ 20x - 7 = A(x - 1) + Bx \]

Expand and collect like terms: \[ 20x - 7 = Ax - A + Bx \] \[ 20x - 7 = (A + B)x - A \]

Equate coefficients:

$ A + B = 20 $
$-A = -7$

From equation 2, solve for $A$: \[ A = 7 \]

Substitute $A = 7$ into equation 1: \[ 7 + B = 20 \] \[ B = 13 \]

Step 4: Write the Decomposition

Substitute $A$ and $B$ back into the partial fractions: \[ \frac{20x - 7}{x(x - 1)} = \frac{7}{x} + \frac{13}{x - 1} \]

Final Answer

The partial fraction decomposition is: \[ \boxed{\frac{7}{x} + \frac{13}{x - 1}} \]

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