Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 69a

Chapter 2, Problem 69a

Find each product. Write answers in standard form. 3i(2-i)²

Verified step by step guidance

First, recognize that the problem requires you to find the product of \$3i$ and the square of the complex number $(2 - i)$, which is written as $3i(2 - i)^2$.

Next, expand the square of the binomial $(2 - i)^2$ using the formula for the square of a binomial: $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = 2$ and $b = i$, so calculate $(2 - i)^2 = 2^2 - 2 \cdot 2 \cdot i + i^2$.

Simplify the expression inside the parentheses by calculating each term: $2^2 = 4$, $-2 \cdot 2 \cdot i = -4i$, and $i^2 = -1$ (since $i$ is the imaginary unit). So, $(2 - i)^2 = 4 - 4i - 1$.

Combine like terms inside the parentheses to get the simplified form of $(2 - i)^2$, which is $3 - 4i$.

Finally, multiply \$3i$ by the simplified expression $(3 - 4i)$ using the distributive property: \(3i \times 3 + 3i \times (-4i)$. Simplify each product, remembering that \)i^2 = -1$, and write the result in standard form $a + bi$.

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Key Concepts

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

Complex Numbers and Imaginary Unit

Complex numbers include a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to manipulate i is essential for working with expressions involving complex numbers.

Exponentiation of Binomials

Raising a binomial to a power, such as (2 - i)², involves applying the distributive property or using the formula (a - b)² = a² - 2ab + b². This step simplifies the expression before further multiplication.

Multiplication of Complex Numbers

Multiplying complex numbers requires distributing each term and combining like terms, remembering to replace i² with -1. This process converts the product into standard form a + bi.

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