Problem 75 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \frac{4}{(2+i)(3-i)} $$

Step-by-Step Solution

Answer

The simplified expression in standard form will be the result of the operation.

Step 1: Compute Denominator’s Conjugate

The conjugate of a complex number \(a+bi\) is \(a-bi\), and the conjugate of \(a-bi\) is \(a+bi\). For the denominator, the complex numbers are \(2+i\) and \(3-i\). Therefore, the conjugates are \(2-i\) and \(3+i\) respectively.

Step 2: Clear the Denominator

Multiply the numerator and denominator by the conjugate of the denominator. Also, remember the FOIL method (First, Outside, Inside, Last) when multiplying two binomials. Hence,\[ \frac{4}{(2+i)(3-i)}\] becomes \[ \frac{4(2-i)(3+i)} {(2+i)(2-i)(3-i)(3+i)}.\] The multiplication of a complex number with its conjugate results in the square of the magnitude of the number, which is a real number.

Step 3: Simplify The Expression

Simplify the equation further, and compute the real and imaginary parts separately. After simplifying, the expression will be in standard form \(a+bi\) where \(a\) and \(b\) will be real numbers.