Problem 3

Question

(a) The complex conjugate of $3+4 i$ is $\overline{3+4 i}=$ _______________ . (b)(3+4 i)(3+4 i)= _________________.

Step-by-Step Solution

Verified

Answer

(a) 3-4i; (b) -7+24i.

1Step 1: Understanding Complex Conjugate

The complex conjugate of a complex number changes the sign of the imaginary part. Therefore, for a complex number in the form of $a + bi$, its conjugate is $a - bi$. The given complex number is $3 + 4i$, so its complex conjugate is $3 - 4i$.

2Step 2: Multiply Complex Numbers

To calculate $(3 + 4i)(3 + 4i)$, use the distributive property (also known as FOIL method for binomials):$\begin{align*}(3 + 4i)(3 + 4i) &= (3)(3) + (3)(4i) + (4i)(3) + (4i)(4i) \&= 9 + 12i + 12i + 16i^2.\end{align*}$We remember that $i^2 = -1$, so this becomes:$9 + 24i + 16(-1) = 9 + 24i - 16 = -7 + 24i$.

Key Concepts

Complex ConjugateImaginary NumbersMultiplying Complex Numbers

Complex Conjugate

A complex conjugate is a fundamental concept in the realm of complex numbers. It's formed by changing the sign of the imaginary part of a complex number. Imagine you're given a complex number such as $3 + 4i$. To find its complex conjugate, simply switch the sign of the term with $i$. This means the conjugate of $3 + 4i$ is $3 - 4i$.

If you have a complex number $a + bi$, its conjugate is $a - bi$.
The complex conjugate mirrors the original number over the real axis.
When a complex number is multiplied by its conjugate, the result is a real number.

This makes the concept incredibly useful, especially in simplifying expressions and finding the magnitude of complex numbers.

Imaginary Numbers

Imaginary numbers are a fascinating component of complex numbers. They're sometimes quite mystical at first glance, but they're simply numbers that involve the imaginary unit "i". The imaginary unit $i$ is defined by the property $i^2 = -1$. Thus, an imaginary number can be thought of as a real number multiplied by the imaginary unit $i$.

For example, the number $4i$ is an imaginary number.
Imaginary numbers create a new dimension of numbers, extending the real line into the complex plane.
In the complex plane, the horizontal line represents real numbers, while the vertical line represents imaginary numbers.

Remember that although these numbers are called "imaginary", they are very real in terms of their utility and applications, especially in engineering and physics.

Multiplying Complex Numbers

Multiplying complex numbers is like dealing with binomials, and it's straightforward once you get the hang of it. The process involves using the distributive property, often referred to as the FOIL (First, Outer, Inner, Last) method.
For instance, consider multiplying two identical complex numbers: $(3 + 4i)(3 + 4i)$. Here's how you do it:

Multiply the first terms: $3 \, \times \, 3 = 9$.
Multiply the outer terms: $3 \, \times \, 4i = 12i$.
Multiply the inner terms: $4i \, \times \, 3 = 12i$.
Multiply the last terms: $4i \, \times \, 4i = 16i^2$.

Combine these results: you'll have $9 + 12i + 12i + 16i^2$. Since $i^2 = -1$, substitute $-1$ for $i^2$ to get $9 + 24i - 16$, which simplifies to $-7 + 24i$.
This method is essential in complex arithmetic, enabling you to simplify expressions and solve equations involving complex numbers.

Problem 3

Other exercises in this chapter

Problem 3

The following questions are about the rational function $$ r(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function $r$ has $x$ -intercepts __________ and _______

View solution

Problem 3

A polynomial of degree $n \geq 1$ has exactly ____ zeros if a zero of multiplicity $m$ is counted $m$ times.

View solution

Problem 3

True or false? If $c$ is a real zero of the polynomial $P,$ then all the other zeros of $P$ are zeros of $P(x) /(x-c)$

View solution

Problem 3

Two polynomials $P$ and $D$ are given. Use either synthetic or long division to divide $P(x)$ by $D(x),$ and express $P$ in the form \(P(x)=D(x) \cdot

View solution