In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (4 - i)² - (1 + 2i)²

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Step 1: Recall the formula for squaring a binomial: $(a + b)^2 = a^2 + 2ab + b^2$. Apply this formula to $(4 - i)^2$, where $a = 4$ and $b = -i$. Expand it as $4^2 + 2(4)(-i) + (-i)^2$.

Step 2: Similarly, apply the same formula to $(1 + 2i)^2$, where $a = 1$ and $b = 2i$. Expand it as $1^2 + 2(1)(2i) + (2i)^2$.

Step 3: Simplify each term in the expansions. For $(4 - i)^2$, calculate $4^2 = 16$, $2(4)(-i) = -8i$, and $(-i)^2 = -1$. Combine these to get $16 - 8i - 1$. For $(1 + 2i)^2$, calculate $1^2 = 1$, $2(1)(2i) = 4i$, and $(2i)^2 = -4$. Combine these to get $1 + 4i - 4$.

Step 4: Subtract the result of $(1 + 2i)^2$ from the result of $(4 - i)^2$. Write this as $(16 - 8i - 1) - (1 + 4i - 4)$. Distribute the negative sign across the second set of parentheses.

Step 5: Combine like terms (real parts and imaginary parts) from the subtraction in Step 4. Simplify the expression to write the result in standard form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding how to manipulate complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. When performing operations on complex numbers, the result should be expressed in this form to clearly identify the real and imaginary components. This is particularly important in problems involving addition, subtraction, or multiplication of complex numbers.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. The formula for binomial expansion involves using the binomial coefficients, which can be found in Pascal's triangle. In the context of complex numbers, this concept is crucial for squaring binomials like (4 - i)^2 and (1 + 2i)^2 to simplify the expression before combining like terms.

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