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Magazine Chapter 8: Problem 18
Complex number expressions can be multiplied using the distributive property or the FOIL technique. Multiply and simplify the following. ( Note: \(i^{2}=-1 .\) ) a. \((3+2 i)(-2+3 i)\) d. \((5-3 i)(5+3 i)\) b. \((4-2 i)(3+i)\) e. \((3-i)^{2}\) c. \((2+i)(2-i)\) f. \((4+5 i)^{2}\)
Short Answer
Expert verified
a. -12 + 5ib. 14 - 2ic. 5d. 34e. 8 - 6if. -9 + 40i
Step by step solution
01
Expand Using Distributive Property
Use the distributive property (FOIL technique) to multiply the terms: \((3 + 2i)(-2 + 3i)\).
02
Apply Distributive Property
Calculate: \(3(-2) + 3(3i) + 2i(-2) + 2i(3i)\)
03
Combine Like Terms
This simplifies to \(-6 + 9i - 4i + 6i^2\). Remember that \(i^2 = -1\), so \(6i^2 = 6(-1) = -6\). Thus, \(-6 + 9i - 4i - 6\).
04
Final Simplification
Combine all real and imaginary parts: \(-12 + 5i\).
05
Expansion of (5-3i)(5+3i)
Use the distributive property for \((5-3i)(5+3i)\), calculated as \(5(5) + 5(3i) - 3i(5) - 3i(3i)\).
06
Simplify
This gives \(25 + 15i - 15i - 9i^2\), and since \(i^2 = -1\), \(-9i^2 = 9\). So, \(25 + 9 = 34\).
07
Expansion of (4-2i)(3+i)
Multiply: \((4-2i)(3+i)\) which becomes \(4(3) + 4(i) - 2i(3) - 2i(i)\).
08
Simplify
This results in \(12 + 4i - 6i - 2(-1)\) because \(i^2 = -1\). Simplifying, this results in \(12 + 2 + (-2i) = 14 - 2i\).
09
Expansion of (2+i)(2-i)
Multiply: \((2+i)(2-i)\) to get \(2(2) + 2(-i) + i(2) - i(i)\).
10
Simplify
\(4 - 2i + 2i - i^2\) simplifies to \(4 - (-1)\) which equals \(5\).
11
Expansion of (3-i)虏
Use the formula \((a-b)^2 = a^2 - 2ab + b^2\): \((3-i)^2 = 3^2 - 2(3)(i) + (-i)^2\).
12
Simplify
\(9 - 6i + (-1)\) resulting in \(8 - 6i\).
13
Expansion of (4+5i)虏
Use the formula \((a+b)^2 = a^2 + 2ab + b^2\): \((4+5i)^2 = 4^2 + 2(4)(5i) + (5i)^2\).
14
Simplify
\(16 + 40i + 25(-1)\) simplifies to \(-9 + 40i\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
distributive property
The distributive property is a useful algebraic rule that allows us to break down complex expressions and simplify multiplication. In the context of complex numbers, applying the distributive property means multiplying every part of the first complex number by every part of the second complex number.
For example, if we need to multiply \( (3 + 2i)(-2 + 3i) \), we distribute each term of the first complex number over each term of the second complex number:
\[ 3(-2) + 3(3i) + 2i(-2) + 2i(3i) \].
By calculating each term step-by-step, we get:
\[-6 + 9i - 4i + 6i^2 \]. Next, we substitute \(i^2 = -1 \), simplifying to \(-6 + 9i - 4i - 6 = -12 + 5i \). By understanding the distributive property, you can handle more complicated expressions by systematically breaking them down into smaller parts.
For example, if we need to multiply \( (3 + 2i)(-2 + 3i) \), we distribute each term of the first complex number over each term of the second complex number:
\[ 3(-2) + 3(3i) + 2i(-2) + 2i(3i) \].
By calculating each term step-by-step, we get:
\[-6 + 9i - 4i + 6i^2 \]. Next, we substitute \(i^2 = -1 \), simplifying to \(-6 + 9i - 4i - 6 = -12 + 5i \). By understanding the distributive property, you can handle more complicated expressions by systematically breaking them down into smaller parts.
FOIL technique
The FOIL technique is a special case of the distributive property used specifically for multiplying two binomials (expressions with two terms each). FOIL stands for First, Outside, Inside, Last, referring to the position of each term in the binomial.
For instance, consider multiplying \( (4 - 2i)(3 + i) \). Using the FOIL method:
\( 12 + 4i - 6i - 2i^2 \). Recognizing that \( i^2 = -1 \), this simplifies to
\( 12 + 4i - 6i + 2 = 14 - 2i \). Mastering the FOIL method helps to quickly and accurately multiply complex binomials.
For instance, consider multiplying \( (4 - 2i)(3 + i) \). Using the FOIL method:
- First: Multiply the first terms: \( 4 \times 3 = 12 \)
- Outside: Multiply the outer terms: \( 4 \times i = 4i \)
- Inside: Multiply the inside terms: \( -2i \times 3 = -6i \)
- Last: Multiply the last terms: \( -2i \times i = -2i^2 \)
\( 12 + 4i - 6i - 2i^2 \). Recognizing that \( i^2 = -1 \), this simplifies to
\( 12 + 4i - 6i + 2 = 14 - 2i \). Mastering the FOIL method helps to quickly and accurately multiply complex binomials.
imaginary unit
The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This definition allows us to extend the number system into the complex plane and solve equations that have no real solutions.
For example, when faced with multiplying \( (5 - 3i)(5 + 3i) \), using the distributive property or FOIL, we get terms containing \(i^2\):
\[ 5(5) + 5(3i) - 3i(5) - 3i(3i) = 25 + 15i - 15i - 9i^2 \].
Since \(i^2 = -1\), we replace \(-9i^2\) with \(+9\), resulting in a final simplified form:
\( 25 + 9 = 34 \). Understanding the imaginary unit is key to simplifying and multiplying complex numbers accurately. It allows us to transition from real parts to imaginary parts and back, leading to a concise solution.
For example, when faced with multiplying \( (5 - 3i)(5 + 3i) \), using the distributive property or FOIL, we get terms containing \(i^2\):
\[ 5(5) + 5(3i) - 3i(5) - 3i(3i) = 25 + 15i - 15i - 9i^2 \].
Since \(i^2 = -1\), we replace \(-9i^2\) with \(+9\), resulting in a final simplified form:
\( 25 + 9 = 34 \). Understanding the imaginary unit is key to simplifying and multiplying complex numbers accurately. It allows us to transition from real parts to imaginary parts and back, leading to a concise solution.
