Problem 62 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$(-5-i)(5+i)$$

Step-by-Step Solution

Answer

The product is \(-24 - 10i\).

Step 1: Distribute the Terms

Use the distributive property (also known as the FOIL method for binomials) to expand the expression \[(-5-i)(5+i).\]Multiply each term in the first binomial by each term in the second binomial:1. \(-5 \times 5 = -25\)2. \(-5 \times i = -5i\)3. \(-i \times 5 = -5i\)4. \(-i \times i = -i^2\)

Step 2: Simplify the Expression

Combine the results from Step 1:Combine like terms and apply the fact that \(i^2 = -1\):\[-25 - 5i - 5i - i^2.\]

Step 3: Substitute \(i^2\) with \(-1\)

Recognize that \(i^2 = -1\), so replace \(-i^2\) with \(-(-1)\):\[-25 - 10i + 1.\]

Step 4: Combine Like Terms

Add the real parts and simplify the expression:\[-25 + 1 = -24.\]Thus, the combined expression is \[-24 - 10i.\]This is the expression in standard form.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions by distributing multiplication over addition or subtraction. When working with binomials, this property is essential.
For example, in the expression \[(-5-i)(5+i),\]the distributive property tells us to multiply each term in the first binomial by each term in the second binomial.

• First, multiply \(-5\) with \(5\), giving us \(-25\).
• Next, multiply \(-5\) with \(i\), producing \(-5i\).
• Then, multiply \(-i\) with \(5\), resulting in \(-5i\) again.
• Finally, multiply \(-i\) with \(i\), leading to \(-i^2\).

By doing this, we successfully expand and simplify the expression using the distributive property.
It ensures each term is appropriately accounted for, making it easier to combine like terms later.

FOIL Method

The FOIL method is a mnemonic that aids in remembering the steps to multiply two binomials. FOIL stands for:

• First - Multiply the first terms in each binomial.
• Outer - Multiply the outer terms in the binomial pair.
• Inner - Multiply the inner terms of the binomials.
• Last - Multiply the last terms in each binomial.

In the expression \[(-5-i)(5+i),\] applying the FOIL method:

• First: \(-5 \times 5 = -25\).
• Outer: \(-5 \times i = -5i\).
• Inner: \(-i \times 5 = -5i\).
• Last: \(-i \times i = -i^2\).

These steps ensure all interactions between terms are captured, just as in the distributive property approach. They give us the same expanded result, which we then simplify further by combining like terms.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a key concept in complex numbers. It is defined by the property that \(i^2 = -1\). This definition allows for the representation of numbers that can extend beyond the real number line.
While working with \(i\), it's essential to remember its unique property which affects calculations within expressions.

• In the expression \(-i^2\), substituting \(i^2\) with \(-1\) gives us \(-(-1)\), simplifying to \(+1\).

This transformation is crucial in converting products involving \(i\) into standard complex form (of \(a + bi\)), as seen in the final result of our example problem. Understanding and applying the imaginary unit correctly allows for accurate solutions when dealing with complex numbers.