Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 11

Chapter 5, Problem 11

Find each product and write the result in standard form. (−5 + 4i)(3 + i)

Verified step by step guidance

Recall that to multiply two complex numbers in the form $(a + bi)(c + di)$, you use the distributive property (FOIL method): multiply each term in the first complex number by each term in the second complex number.

Apply the distributive property: $(−5 + 4i)(3 + i) = (−5)(3) + (−5)(i) + (4i)(3) + (4i)(i)$.

Calculate each product separately: $(−5)(3) = −15$, $(−5)(i) = −5i$, $(4i)(3) = 12i$, and $(4i)(i) = 4i^2$.

Remember that $i^2 = -1$, so replace \$4i^2$ with $4(-1) = -4$.

Combine the real parts and the imaginary parts: real parts are $-15$ and $-4$, imaginary parts are $-5i$ and \$12i$. Write the final expression in the form $a + bi$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Complex Number Multiplication

Multiplying complex numbers involves using the distributive property (FOIL method) to expand the product of two binomials. Each term is multiplied, remembering that i² = -1, which simplifies the expression into a standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary coefficient. After multiplication, the result should be simplified and rearranged to clearly separate real and imaginary parts.

Imaginary Unit Properties

The imaginary unit i is defined such that i² = -1. This property is essential when simplifying products involving i, as it converts powers of i into real numbers, allowing the expression to be written in standard form.

Recommended video:

2:20

Imaginary Roots with the Square Root Property

Related Practice

Textbook Question

In Exercises 11–14, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. 1 − i

467

views

Textbook Question

Indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, −3π/4)

794

views

Textbook Question

Plot each complex number and find its absolute value. z = −3 + 4i

633

views

Textbook Question

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (2, 45°)

823

views

Textbook Question

Perform the indicated operations and write the result in standard form. (4 + √−8 )/ 2

666

views

Textbook Question

Use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x = t − 2, y = 2t + 1; −2 ≤ t ≤ 3

833

views