Problem 63 - Algebra 2

Question

Simplify. $$ (8-i)(4-3 i) $$

Step-by-Step Solution

Answer

The simplified form is \(29 - 28i\).

Step 1: Distribute the First Term

Multiply the first term of the first binomial, which is 8, by both terms of the second binomial, \(4 - 3i\). This gives us \(8 \cdot 4 = 32\) and \(8 \cdot (-3i) = -24i\).

Step 2: Distribute the Second Term

Multiply the second term of the first binomial, which is \(-i\), by both terms of the second binomial \(4 - 3i\). This gives us \(-i \cdot 4 = -4i\) and \(-i \cdot (-3i) = 3i^2\).

Step 3: Simplify the Imaginary Unit

Recall that \(i^2 = -1\). Thus, \(3i^2 = 3(-1) = -3\). This simplifies the previously calculated \(3i^2\) term.

Step 4: Combine Like Terms

Combine all the results obtained in Steps 1 to 3. We have \(32 - 24i - 4i - 3\). Combine the real parts and the imaginary parts: \(32 - 3 = 29\) and \(-24i - 4i = -28i\).

Key Concepts

Understanding Binomials

When working with algebra, a binomial is an expression that contains two terms. Each term is usually separated by a plus or minus sign. For example, in the expression \((8 - i)(4 - 3i)\), there are two binomials: \(8 - i\) and \(4 - 3i\). These expressions are crucial in expanding products and simplifying complex numbers.
To work with binomials effectively, you'll often use the distributive property, which is sometimes remembered by the acronym FOIL (First, Outer, Inner, Last). This helps to multiply each term in the first binomial with each term in the second binomial, ensuring that no term is missed.

The Imaginary Unit

In mathematics, the imaginary unit is denoted by \(i\) and is defined as the square root of \(-1\). The equation \(i^2 = -1\) is fundamental when dealing with complex numbers. This concept allows us to extend the real number system to include numbers that represent the square roots of negative numbers.
When simplifying expressions involving complex numbers, always remember to use the property \(i^2 = -1\) to transform terms with \(i\) squared into real numbers. This simplification helps in combining and reducing terms effectively.

The Process of Simplification

Simplification in mathematics means reducing an expression to its simplest form. This can involve expanding binomials, like in our example, and using the properties of imaginary numbers for further reduction.
After using the distributive property on the binomials, you'll often need to apply the rule \(i^2 = -1\). This crucial step transforms complex expressions into simpler forms. For instance, in the given exercise, the term \(3i^2\) becomes \(-3\), a real number, which is much simpler to work with.

Combining Like Terms

Once you have expanded and simplified the expressions, the next step is to combine like terms. Like terms are terms in the expression that have the same variables raised to the same power.
In our complex number example, real parts (like \(32\) and \(-3\)) and imaginary parts (like \(-24i\) and \(-4i\)) should be combined separately. This yields the simplest form of the expression.

• For real terms: Combine to get the final real part.
• For imaginary terms: Combine to get the final imaginary part.

In our case, combining \(32 - 3\) gives us \(29\), and \(-24i - 4i\) gives us \(-28i\). These steps ensure a clean and comprehensible final result.