To simplify the expression $\frac{1 - 5i}{4 + i}$ into a standard complex form, follow these steps:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator

The conjugate of $4 + i$ is $4 - i$.

$\frac{1 - 5i}{4 + i} \times \frac{4 - i}{4 - i} = \frac{(1 - 5i)(4 - i)}{(4 + i)(4 - i)}$

Step 2: Expand the numerator

$(1 - 5i)(4 - i) = 1 \cdot 4 + 1 \cdot (-i) - 5i \cdot 4 - 5i \cdot (-i)$ $= 4 - i - 20i + 5i^2$ Since $i^2 = -1$, substitute $-1$ for $i^2$: $= 4 - i - 20i + 5(-1) = 4 - i - 20i - 5$ Combine like terms: $= (4 - 5) + (-1 - 20)i = -1 - 21i$

Step 3: Simplify the denominator

$(4 + i)(4 - i) = 4^2 - i^2 = 16 - (-1) = 16 + 1 = 17$

Step 4: Write the expression in standard form

$\frac{1 - 5i}{4 + i} = \frac{-1 - 21i}{17} = -\frac{1}{17} - \frac{21}{17}i$

Final Answer

$\frac{1 - 5i}{4 + i} = -\frac{1}{17} - \frac{21}{17}i$

Would you like further details on any steps?

Here are some additional questions to explore:

1. How do you find the conjugate of a complex number?
2. Why does multiplying by the conjugate help simplify complex fractions?
3. How do you add and subtract real and imaginary parts in complex numbers?
4. What is the standard form of a complex number, and why is it useful?
5. How is the modulus of a complex number calculated?

Tip: Always use the conjugate of the denominator to eliminate imaginary parts in the denominator when simplifying complex fractions.