Methods Resolve the following into Pa. (x−1), (x+2)(x−3). Solution 1

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ScanSolve AI Tutor · Accepted Answer

You're asking for the second method to resolve $\frac{x-1}{(x+2)(x-3)}$ into partial fractions. This method is often called the substitution method or Heaviside cover-up method.

Set up the partial fraction decomposition.
$\frac{x-1}{(x+2)(x-3)} = \frac{A}{x+2} + \frac{B}{x-3}$
Multiply both sides by $(x+2)(x-3)$ to clear the denominators:
$x-1 = A(x-3) + B(x+2)$
Solve for $A$ using the substitution method.
To find $A$ , substitute the value of $x$ that makes the term with $B$ zero. This occurs when $x+2=0$ , so $x=-2$ .
Substitute $x=-2$ into the equation $x-1 = A(x-3) + B(x+2)$ :
$(-2)-1 = A(-2-3) + B(-2+2)$ $-3 = A(-5) + B(0)$ $-3 = -5A$ $A = \frac{-3}{-5}$ $A = \frac{3}{5}$
Solve for $B$ using the substitution method.
To find $B$ , substitute the value of $x$ that makes the term with $A$ zero. This occurs when $x-3=0$ , so $x=3$ .
Substitute $x=3$ into the equation $x-1 = A(x-3) + B(x+2)$ :
$(3)-1 = A(3-3) + B(3+2)$ $2 = A(0) + B(5)$ $2 = 5B$ $B = \frac{2}{5}$
Write the final partial fraction decomposition.
Substitute the values of $A$ and $B$ back into the partial fraction form:
$\frac{x-1}{(x+2)(x-3)} = \frac{\frac{3}{5}}{x+2} + \frac{\frac{2}{5}}{x-3}$
This can be written as:
$\frac{3}{5(x+2)} + \frac{2}{5(x-3)}$

Methods Resolve the following into Pa. (x-1), (x+2)(x-3). Solution 1

Answer

Step-by-step solution

Set up the partial fraction decomposition.

Solve for $A$ using the substitution method.

Solve for $B$ using the substitution method.

Write the final partial fraction decomposition.

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