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Chapter 2: Problem 11

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

Short Answer

Expert verified
The product of \((-5+4i)(3+i)\) in standard form is \(-15-1+7i = -16 + 7i\).

Step by step solution

01

Apply the Distributive Property

Start by applying the distributive property to multiply each term in the first complex number, \(-5+4i\), with each term in the second complex number, \(3+i\). This will give four products: \(-5 \cdot 3, -5 \cdot i, 4i \cdot 3, 4i \cdot i\)
02

Calculate the products

Calculate each of these four products, getting \(-15, -5i, 12i, 4i^2.\) Remember that \(i^2 = -1\) by definition.
03

Simplify the expression

Add the results from Step 2 together and simplify where possible. Specifically, replace \(i^2\) with \(-1\) and combine like terms to get the final answer in standard form, \(a+bi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When working with complex numbers, the distributive property is a crucial tool. It allows us to multiply each term inside one complex number by each term in another. This technique is similar to what we do when multiplying polynomials. In our exercise, we have two complex numbers:
  • First complex number: \(-5 + 4i\)
  • Second complex number: \(3 + i\)
Using the distributive property, each term from the first complex number is multiplied by each term from the second. This results in four separate products:
  • \(-5 \cdot 3\)
  • \(-5 \cdot i\)
  • \(4i \cdot 3\)
  • \(4i \cdot i\)
Once these products are calculated, two important steps follow: combining the like terms and simplifying the expression for the final result.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a cornerstone of complex number arithmetic. It is defined by the property that \(i^2 = -1\).
When working through complex number problems involving products or powers of \(i\), always remember this property. In the given exercise:
  • The term \(4i \cdot i\) becomes \(4i^2\), which simplifies to \(4(-1)\) or \(-4\), using the definition \(i^2 = -1\).
This transformation is key in reducing the complexity of calculations involving imaginary numbers, allowing us to neatly simplify and eventually write results in a standardized format.
Standard Form of a Complex Number
The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. In any problem involving complex numbers, particularly when multiplying, the goal is to express the final result in this form.
After applying the distributive property and simplifying using the imaginary unit \(i\), the step to combine like terms is essential. In our exercise:
  • The products calculated were: \(-15, -5i, 12i,\) and \(-4\).
  • We combine the imaginary terms: \(-5i + 12i = 7i\).
  • We then combine the real parts: \(-15 - 4 = -19\).
Thus, the product expressed in standard form is \(-19 + 7i\). Writing complex numbers in standard form ensures clarity and consistency in mathematical communication.

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