Problem 72 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(8-4 i)(7-2 i)$$

Step-by-Step Solution

Answer

\(48 - 44i\)

Step 1: Apply FOIL

\((8-4i)(7-2i) = 8(7) + 8(-2i) + (-4i)(7) + (-4i)(-2i)\)
\(= 56 - 16i - 28i + 8i^2\)

Step 2: Simplify using i² = -1

\(= 56 - 44i + 8(-1) = 56 - 44i - 8 = 48 - 44i\)

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle that allows us to multiply a single term across terms inside a parenthesis. It can be expressed as:

• If you have any expression of the form \(a(b + c)\), it becomes \(ab + ac\).

When it comes to complex numbers, this property is utilized to simplify expressions even further. Let's first consider what happens when you multiply something like \((8-4i)(7-2i)\):

• The first term \(8\) is multiplied by both \(7\) and \(-2i\), resulting in \(8 \times 7\) and \(8 \times (-2i)\).
• In a similar way, \(-4i\) is multiplied by \(7\) and \(-2i\): so \((-4i) \times 7\) and \((-4i) \times (-2i)\).

This effectively breaks down the math into smaller, more manageable steps. Keep in mind this is not just limited to complex numbers but applied universally across algebra.

FOIL Method

When dealing with the multiplication of two binomials, like \((a + bi)(c + di)\), the FOIL method is a special application of the distributive property. FOIL is an acronym that stands for:

• **First**: Multiply the first terms in each binomial.
• **Outer**: Multiply the outer terms in the binomials.
• **Inner**: Multiply the inner terms.
• **Last**: Multiply the last terms in each binomial.

In our initial problem, \((8-4i)(7-2i)\), the FOIL steps would be:

• First: \(8 \times 7\)
• Outer: \(8 \times (-2i)\)
• Inner: \(-4i \times 7\)
• Last: \(-4i \times (-2i)\)

These give us the necessary components to combine and simplify the ultimate product.

Standard Form of a Complex Number

Complex numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).
Let's see how our problem looks once the multiplications have been worked through:

• First part: \(8 \times 7 = 56\)
• Outer part: \(8 \times -2i = -16i\)
• Inner part: \(-4i \times 7 = -28i\)
• Last part: \(-4i \times -2i = 8i^2\), and since \(i^2 = -1\), this becomes \(-8\)

Now combine all of these:
The real numbers \(56\) and \(-8\) sum to \(48\), while the imaginary parts \(-16i\) and \(-28i\) sum to \(-44i\).
So, \(8-4i)(7-2i)\) in standard form is \(48 - 44i\).
The standard form of a complex number is essential as it provides a consistent way to write complex numbers, making them easy to read and compare.