Problem 22 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{3}{4+i} $$

Step-by-Step Solution

Answer

The final result of the division is \(12/17 - (3/17)i\).

Step 1: Identify the Conjugate

The conjugate of a complex number \(a+bi\) is the number \(a-bi\). Therefore, the conjugate of \(4+i\) is \(4-i\).

Step 2: Multiply by the Conjugate

In order to remove the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator: \((3/(4+i)) * ((4-i)/(4-i))\).

Step 3: Simplify the Numerator and Denominator

When multiplying, apply the distributive property in both the numerator and denominator. In the numerator, we get \(3*(4-i) = 12-3i\). In the denominator, use the formula \((a+b)*(a-b) = a^2 - b^2\), thus we get \((4+i)*(4-i) = 4^2 - (i)^2 = 16 - (-1) = 17\).

Step 4: Write the Answer in Standard Form

So the division becomes \(12/17 - (3/17)i\). This is the standard form of the complex number, where the real part is \(12/17\) and the imaginary part is \(-3/17\). This is the final result.