Question Rewrite $f(x) = \frac{8x}{x^2 - 4x + 3}$ using partial fractions.

Studdy Solution

STEP 1

Assumptions
1. The given function is $f(x) = \frac{8x}{x^2 - 4x + 3}$ .
2. We want to express the function as a sum of partial fractions.
3. The denominator can be factored into linear factors since it is a quadratic polynomial.

STEP 2

Factor the denominator of the function to find the potential partial fractions.
$x^2 - 4x + 3 = (x - a)(x - b)$

STEP 3

Find the roots of the quadratic equation $x^2 - 4x + 3 = 0$ to determine the values of $a$ and $b$ .

STEP 4

Use the quadratic formula to find the roots:
$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

STEP 5

Simplify the expression under the square root and calculate the roots:
$x = \frac{4 \pm \sqrt{16 - 12}}{2}$
$x = \frac{4 \pm \sqrt{4}}{2}$

STEP 6

Finish calculating the roots:
$x = \frac{4 \pm 2}{2}$
$x = 1 \text{ or } x = 3$

STEP 7

Now we can factor the denominator using the roots:
$x^2 - 4x + 3 = (x - 1)(x - 3)$

STEP 8

Express the function as a sum of partial fractions with unknown coefficients $A$ and $B$ :
$\frac{8x}{(x - 1)(x - 3)} = \frac{A}{x - 1} + \frac{B}{x - 3}$

STEP 9

To find the values of $A$ and $B$ , multiply both sides of the equation by the common denominator $(x - 1)(x - 3)$ :
$8x = A(x - 3) + B(x - 1)$

STEP 10

Expand the right side of the equation:
$8x = Ax - 3A + Bx - B$

STEP 11

Combine like terms:
$8x = (A + B)x - (3A + B)$

STEP 12

Since the left side of the equation is just $8x$ , we can equate the coefficients of the corresponding powers of $x$ on both sides of the equation:
For the $x$ term: $A + B = 8$
For the constant term: $-3A - B = 0$

STEP 13

We now have a system of linear equations to solve for $A$ and $B$ :
\begin{align} A + B &= 8 \\ -3A - B &= 0 \end{align}

STEP 14

Solve the system of equations. Start by adding the second equation to the first to eliminate $B$ :
\begin{align} A + B &= 8 \\ (-3A - B) + (A + B) &= 0 + 8 \end{align}

STEP 15

Simplify the resulting equation:
\begin{align} -2A &= 8 \\ A &= -4 \end{align}

STEP 16

Substitute $A = -4$ into the first equation to find $B$ :
\begin{align} -4 + B &= 8 \\ B &= 8 + 4 \\ B &= 12 \end{align}

STEP 17

Now that we have the values of $A$ and $B$ , we can write the original function as a sum of partial fractions:
$f(x) = \frac{-4}{x - 1} + \frac{12}{x - 3}$
This is the expression of the function using partial fractions.

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