Problem 30 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$(-2+i)(3-7 i)$$

Step-by-Step Solution

Answer

The expression evaluates to \(1 + 17i\).

Step 1: Apply the Distributive Property

To evaluate the expression \[(-2+i)(3-7i)\]start by applying the distributive property (also known as the FOIL method for binomials):1. Multiply the first terms: \(-2 \times 3 = -6\) 2. Multiply the outer terms: \(-2 \times -7i = 14i\) 3. Multiply the inner terms: \(i \times 3 = 3i\) 4. Multiply the last terms: \(i \times -7i = -7i^2\)

Step 2: Simplify the Expression

Now, add all the results from the first step:\[-6 + 14i + 3i - 7i^2\]Note that \(i^2 = -1\), so simplify \(-7i^2\) to \(7\). The expression becomes:\[-6 + 14i + 3i + 7\]

Step 3: Combine Like Terms

Combine the real and imaginary parts:Real part: \(-6 + 7 = 1\)Imaginary part: \(14i + 3i = 17i\)Thus, the expression simplifies to:\[1 + 17i\]

Step 4: Present the Answer

The expression \((-2+i)(3-7i)\) evaluates to:\[1 + 17i\] This is in the form of \(a + bi\), where \(a = 1\) and \(b = 17\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps simplify expressions, particularly when dealing with printense terms, bins, or more complex equations. When applied to multiplication over addition or subtraction, this property allows us to multiply each term inside the parentheses by the factor outside.

This technique is essential when working with binomials like \((-2+i)\) and \((3-7i)\). When you apply the distributive property to these two binomials, you're essentially using the FOIL (First, Outer, Inner, Last) method:

• First: Multiply the first terms of each binomial, i.e., \(-2\cdot 3 = -6\).
• Outer: Multiply the outer terms, i.e., \(-2\cdot -7i = 14i\).
• Inner: Multiply the inner terms, i.e., \(i\cdot 3 = 3i\).
• Last: Multiply the last terms, i.e., \(i\cdot -7i = -7i^2\).

Once all these products are obtained, they are added together to complete the expression. This simplification will make the equation easier to solve and understand.

Remember, correct application of the distributive property is key to tackling complex numbers and other algebraic expressions.

Imaginary Unit

In the realm of complex numbers, the imaginary unit, denoted by \(i\), is a critical component. The imaginary unit is defined as the square root of -1, that is, \(i^2 = -1\). This unique property allows us to work with numbers outside the real number system.

When dealing with complex multiplication, such as with \((-7i^2)\), knowing that \(i^2 = -1\) becomes invaluable. By substituting -1 for \(i^2\), we can further simplify expressions that would otherwise seem abstract or unsolvable.

Consider the expression from our problem:

• The term \(-7i^2\) can be rewritten and simplified by using the property of \(i\). Since \(i^2 = -1\), we replace \(-7i^2\) with 7, converting an imaginary expression into a real number.

This transformation, facilitated by the imaginary unit, allows real-valued simplification and subsequent combinative arithmetic to occur, resulting in solutions comprehensible in the complex plane.

Binomials

A binomial is a type of polynomial that consists of exactly two terms. These terms are typically separated by either a plus or a minus sign. In complex number arithmetic, binomials can involve both real and imaginary components.

For instance, in \((-2+i)\) and \((3-7i)\), each binomial contains:

• A real part: \(-2\) and \(3\), respectively.
• An imaginary part: \(i\) and \(-7i\), respectively.

The operations carried out between binomials involve applying properties such as distribution and understanding how real and imaginary parts interact. When multiplying two binomials,

each term of the first binomial is multiplied with each term of the second binomial, utilizing the distributive property to ensure no part is missed. This results in four distinct products, each with either an entirely real outcome, an entirely imaginary outcome, or a mix of both.

The final step in solving binomial expressions involving complex numbers is to combine like terms:

• Join real components.
• Join imaginary components.

This results in an expression of the form \(a+bi\), easily interpretable and critical for further mathematical operations or transformations.