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Chapter 8: Problem 34

In Exercises 13-40, perform the indicated operation, simplify, and express in standard form. $$ (-3-2 i)(7-4 i) $$

Short Answer

Expert verified
The expression is -29 - 2i in standard form.

Step by step solution

01

Apply the Distributive Property

To find the product (-3 - 2i)(7 - 4i), use the distributive property (FOIL method for two binomials). Distribute each term in the first binomial by each term in the second one: First: (-3)(7) = -21 Outside: (-3)(-4i) = 12i Inside: (-2i)(7) = -14i Last: (-2i)(-4i) = 8i^2.
02

Combine Like Terms

Now, combine all terms from the distribution: -21 + 12i - 14i + 8i^2. Combine the imaginary parts: 12i - 14i = -2i.
03

Simplify Using i虏 = -1

Remember that i虏 = -1, so substitute this into the expression: 8i虏 = 8(-1) = -8. Now the expression becomes: -21 - 2i - 8.
04

Final Simplification and Standard Form

Combine the real numbers: -21 - 8 = -29. Thus, the expression simplifies to: -29 - 2i.

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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 fundamental algebraic concept that helps in multiplying a single term by two or more terms inside a set of parentheses. When it comes to complex numbers, this property is crucial, especially in multiplication operations involving binomials.
  • In the context of complex numbers multiplication, the distributive property allows us to multiply each term in the first binomial by each term in the second binomial.
  • This ensures that every combination of terms is considered, making the multiplication complete and systematic.
This is particularly helpful when simplifying expressions, ensuring that no part of the expression is left unaccounted for. It is like opening up each pair of parentheses and extracting every possible multiplication combination between the sets, which then can be further combined and simplified.
Imaginary Numbers
Imaginary numbers are numbers that involve the square root of negative one, represented by the symbol \( i \). In many calculations, especially those involving complex numbers, \( i \) plays a crucial role.
  • Imaginary numbers arise when you take the square root of a negative number, something that is impossible in the set of real numbers.
  • In our exercise, the presence of terms like \(-3 - 2i\) and \(-4i\) indicate the imaginary components within the complex numbers.
When simplifying expressions with imaginary numbers, it is important to remember that \( i^2 = -1 \). This allows for converting squared imaginary numbers back into real numbers, aiding the simplification process and eventually expressing the final result in standard form.
Standard Form Simplification
Standard form for complex numbers is important to follow because it arranges the number in a way that makes it easily understandable. In standard form, a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers.
  • After performing all operations, always rewrite the expression in this \( a + bi \) format.
  • This includes collecting all real components under \( a \) and placing all imaginary components under \( bi \) for clarity.
This standardization ensures that your answer is easily recognizable and adheres to mathematical conventions, making communication of complex numbers straightforward and eliminating ambiguity.
FOIL Method
The FOIL method is a specific application of the distributive property for binomials, an effective way to multiply two binomials. It stands for First, Outside, Inside, Last, referring to the terms that initially get multiplied.
  • First: Multiply the first terms in each binomial.
  • Outside: Multiply the outer terms in the product.
  • Inside: Multiply the inner terms.
  • Last: Multiply the last terms in each binomial.
This method makes it easy to remember and apply the distributive property specifically to two-term expressions. When using the FOIL method in complex numbers, it facilitates the combination of real and imaginary parts, ensuring a clear path to simplification and further manipulation.

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Most popular questions from this chapter

In Exercises 79 and 80, explain the mistake that is made. Convert the rectangular coordinate \((-2,-2)\) to polar coordinates. Solution: Label \(x\) and \(y . \quad x=-2, y=-2\) Find \(r . \quad r=\sqrt{x^{2}+y^{2}}=\sqrt{4+4}=\sqrt{8}=2 \sqrt{2}\) Find \(\theta . \quad \tan \theta=\frac{-2}{-2}=1\) $$ \theta=\tan ^{-1}(1)=\frac{\pi}{4} $$ Write the point in polar coordinates. \(\left(2 \sqrt{2}, \frac{\pi}{4}\right)\) This is incorrect. What mistake was made?

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=4\left(t^{2}+1\right), y=1-t^{2} $$

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r=2+3 \sin \theta $$

For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Flight of a Projectile. A projectile is launched from the ground at a speed of 400 feet per second at an angle of \(45^{\circ}\) with the horizontal. How far does the projectile travel (what is the horizontal distance), and what is its maximum altitude? (Note the symmetry of the projectile path.)

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=4 \cos (2 t), y=t, t \text { in }[0,2 \pi] $$
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