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

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

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

Problema 11

Capitolo 5, Problema 11

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

Guida verificata passo dopo passo

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 $a + bi$.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.

Video consigliato:

Percorso guidato

5:02

Multiplying Complex Numbers

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.

Video consigliato:

Percorso guidato

04:47

Complex Numbers In Polar Form

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.

Video consigliato:

Percorso guidato

2:20

Imaginary Roots with the Square Root Property

Pratica correlata

Domanda del libro di testo

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

Domanda del libro di testo

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

Domanda del libro di testo

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

633

views

Domanda del libro di testo

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

Domanda del libro di testo

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

666

views

Domanda del libro di testo

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