91Ó°ÊÓ

The distributive property is a fundamental algebraic property used to simplify expressions. It states that for any numbers or terms, multiplying a single term by a sum of terms is the same as multiplying each term individually and then adding the results.

In the context of our exercise, we applied the distributive property to multiply \(-4i\) by each term in the binomial \(3 - i\). This broke down the multiplication into two simpler parts: \(-4i \times 3\) and \(-4i \times (-i)\).

Remember, by distributing the term across the binomial, we can handle complex products step by step. This simplifies the problem and prevents mistakes.

The imaginary unit, denoted as \(i\), is a special number defined by the property that \(i^2 = -1\). This concept is crucial when working with complex numbers.

In the step-by-step solution provided, we encountered \(4i^2\). Knowing that \(i^2 = -1\) allowed us to simplify \(4i^2\) into \(4 \times (-1) = -4\).

Understanding the imaginary unit helps us remember:

• \(i = \sqrt{-1}\)
• \(i^2 = -1\)
• \(i^3 = -i \)
• \(i^4 = 1\)

Each subsequent power of \(i\) cycles through these values. Recognizing these properties makes working with complex numbers more manageable.

A binomial is a polynomial with exactly two terms, separated by either a plus \( + \) or minus \(- \) sign. In our problem, we worked with the binomial \(3 - i\).

When multiplying a binomial by another term, like \(-4i\) in this example, using the distributive property simplifies the process. Breaking the multiplication into steps, as we did, helps in managing each part of the binomial separately.

In more complex cases, such as multiplying two binomials, the distributive property extends into the FOIL method (First, Outer, Inner, Last) to ensure every term is correctly multiplied. For instance, if we were to multiply \((a + b)\) and \((c + d)\), we would calculate \((a \times c, a \times d, b \times c, b \times d)\) and then sum up the results.

Whether handling simple or more complex multiplications, mastering binomials is essential for success in algebra and beyond.