Questions: Use partial fraction decomposition to solve for constants A and B: (2x+1)/(x^2-x-20)=(A)/(x+4)+(B)/(x-5). A=7 / 2 B= Adomterialnation, integer or exact decimal. Do not approximate.

Solution Steps

To solve for the constants \( A \) and \( B \) in the partial fraction decomposition \(\frac{2x+1}{x^2-x-20} = \frac{A}{x+4} + \frac{B}{x-5}\), follow these steps:

1. Factor the denominator \( x^2 - x - 20 \) into \((x+4)(x-5)\).
2. Set up the equation \(\frac{2x+1}{(x+4)(x-5)} = \frac{A}{x+4} + \frac{B}{x-5}\).
3. Multiply both sides by \((x+4)(x-5)\) to clear the denominators.
4. Expand and collect like terms to form a polynomial equation.
5. Equate the coefficients of corresponding powers of \( x \) on both sides to form a system of linear equations.
6. Solve the system of equations to find \( A \) and \( B \).

First, we need to factor the denominator of the given fraction: \[ x^2 - x - 20 \] We look for two numbers that multiply to \(-20\) and add to \(-1\). These numbers are \(4\) and \(-5\). Therefore, we can factor the quadratic as: \[ x^2 - x - 20 = (x + 4)(x - 5) \]

We are given the equation: \[ \frac{2x + 1}{x^2 - x - 20} = \frac{A}{x + 4} + \frac{B}{x - 5} \] Substituting the factored form of the denominator, we get: \[ \frac{2x + 1}{(x + 4)(x - 5)} = \frac{A}{x + 4} + \frac{B}{x - 5} \]

To combine the fractions on the right-hand side, we need a common denominator: \[ \frac{A}{x + 4} + \frac{B}{x - 5} = \frac{A(x - 5) + B(x + 4)}{(x + 4)(x - 5)} \] Equating the numerators, we get: \[ 2x + 1 = A(x - 5) + B(x + 4) \]

Expand the right-hand side: \[ 2x + 1 = Ax - 5A + Bx + 4B \] Combine like terms: \[ 2x + 1 = (A + B)x + (4B - 5A) \]

By comparing coefficients, we get the following system of equations: \[ A + B = 2 \] \[ 4B - 5A = 1 \]

We are given \(A = \frac{7}{2}\). Substitute \(A\) into the first equation: \[ \frac{7}{2} + B = 2 \] Solve for \(B\): \[ B = 2 - \frac{7}{2} = \frac{4}{2} - \frac{7}{2} = -\frac{3}{2} \]

Final Answer