Problem 71 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(6-2 i)(7-i)$$

Step-by-Step Solution

Answer

The product is \(40 - 20i\).

Step 1: Distribute the Terms

To find the product of two complex numbers, distribute each term in the first complex number across each term in the second complex number. So, compute: \((6-2i)(7-i) = 6 \times 7 + 6 \times (-i) + (-2i) \times 7 + (-2i) \times (-i)\).

Step 2: Multiply the Real and Imaginary Parts

Perform the multiplication of each term: \(6 \times 7 = 42\), \(6 \times (-i) = -6i\), \(-2i \times 7 = -14i\), \(-2i \times (-i) = 2i^2\).

Step 3: Simplify Using \(i^2 = -1\)

Since \(i^2 = -1\), replace \(i^2\) with \(-1\) in the product term: \(2i^2 = 2 \times (-1) = -2\).

Step 4: Combine Like Terms

Now, combine all terms from step 2 and incorporate the simplified \(i^2\) result from step 3: \(42 - 6i - 14i - 2\).

Step 5: Finalize the Complex Number

Add the real parts together and the imaginary parts together: Real part: \(42 - 2 = 40\), Imaginary part: \(-6i - 14i = -20i\). Thus, the complex number in standard form is \(40 - 20i\).

Key Concepts

Multiplying Complex Numbers

Complex numbers are fascinating entities that broaden our understanding of algebra beyond real numbers. A complex number typically has the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is an imaginary unit defined by \(i^2 = -1\). Multiplying complex numbers involves more than just straightforward multiplication, due to the presence of imaginary parts.

• First, apply the distributive property, which is a fundamental algebraic principle.


• Next, multiply each term in one complex number by every term in the other one.

For example, when multiplying \((6 - 2i)\) by \((7 - i)\):

• Distribute \(6\) across both \(7\) and \(-i\).
• Do the same for \(-2i\), multiplying it by both terms in \(7 - i\).

This step follows a similar process to the distribution of terms when multiplying binomials in traditional algebra, with the additional step of handling the imaginary unit \(i\).

Standard Form of a Complex Number

The standard form of a complex number is succinctly written as \(a + bi\), where \(a\) is the real part, and \(b\) is the coefficient of the imaginary part. After performing operations on the complex number, it's crucial to ensure the final expression is in this form.

• The real part consists of all non-imaginary terms combined together.


• The imaginary part is consolidated under a single imaginary unit \(i\).

For instance, from our multiplication, we obtained terms like \(42, -6i, -14i,\) and \(-2\). To express them in the standard form:

• Add up all the real parts: \(42 - 2 = 40\).


• Combine all imaginary terms: \(-6i - 14i = -20i\).

The resulting standard form becomes \(40 - 20i\).

Distribution in Algebra

Distribution is a fundamental concept in algebra that helps simplify expressions and solve equations efficiently. It involves spreading, or distributing, each term across the terms within a parenthesis.

• The distributive property states \(a(b + c) = ab + ac\).


• This property applies equally to complex numbers and polynomials.

In the operation \((6-2i)(7-i)\), the distribution process looks like this:

• You multiply \(6\) by both \(7\) and \(-i\).
• Then, \(-2i\) is multiplied with both \(7\) and \(-i\).

The concept hinges on breaking down and handling each part systematically, ensuring increased comprehensiveness and accuracy for broader algebraic expressions in complex number multiplicities. By mastering distribution with complex numbers, you deepen your algebraic problem-solving abilities.