Question

Describe and correct the error in performing the operation and writing the answer in standard form. $$ \begin{aligned} (3+2 i)(5-i) &=15-3 i+10 i-2 i^2 \\ &=15+7 i-2 i^2 \\ &=-2 i^2+7 i+15 \end{aligned} $$

Short answer

The error in the operation was that the student did not correctly apply the identity \(i^2 = -1\). After applying the identity and simplification, the correct standard form is \(17+7i\).

Step-by-step solution

Analyze the student's calculations

In these calculations, the student has started off correctly by expanding the brackets: multiplying every term in the first bracket by every term in the second one. However, the mistake lies in the third line where the student had written \(-2i^2\). The correct identity for \(i^2\) was not applied correctly. Instead of substituting \(-1\) for \(i^2\), the student has left it as \(i^2\).

Apply Identity \(\$i^2 = -1\$

The identity \(i^2 = -1\) should have been applied. This means that wherever \(i^2\) appears, it should be replaced with \(-1\). Which leads to: \(15+7i -2(-1) = 15 +7i + 2\).

Simplify further

By further simplification, \(15 +7i + 2\) simplifies to \(17+7i\). This is the standard form of the complex number in the question

Key concepts

complex number operations

Complex numbers expand our understanding of arithmetic, offering a way to handle numbers involving the square root of negative one. To perform operations with complex numbers, we typically treat them like binomials. For example, in the given exercise, we have two complex numbers, \( (3 + 2i) \) and \( (5 - i) \), which are multiplied together. This process is similar to multiplying binomials with real numbers.

Firstly, each term in the first complex number is multiplied by each term in the second.

• 3 multiplied by 5 equals 15.
• 3 multiplied by \(-i\) gives \(-3i\).
• 2i multiplied by 5 results in \(10i\).
• 2i multiplied by \(-i\) delivers \(-2i^2\).

This operation follows the distributive property, allowing for the reorganization and collection of like terms. By correctly executing these operations, you'll find correct results step by step.

standard form of complex numbers

The standard form of a complex number is expressed as \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. This form helps in clearly distinguishing and manipulating complex numbers.

In the exercise above, after performing the multiplication correctly and applying the identities, the resulting expression is correctly converted into the standard form of \( 17 + 7i \). It’s crucial to rearrange the terms in descending order of their parts—real first, imaginary second—to follow the standard form. This practice brings clarity and uniformity to complex number expressions across calculations and makes operations more straightforward.

imaginary unit i

The imaginary unit \( i \) is defined as the square root of \(-1\). This is a fundamental concept in complex numbers, acting as a building block of the imaginary component within complex expressions. Understanding \( i \) allows us to extend our number system to include solutions to equations like \( x^2 = -1 \).

In operations with complex numbers, an important identity used is \( i^2 = -1 \). Applying this identity properly is pivotal in solving complex number problems, as seen in the exercise where there was an initial error. Failing to substitute \(-1\) for \( i^2 \) can lead to incorrect results.

• Always replace \( i^2 \) with \(-1\).
• Carefully simplify your results afterwards to ensure accuracy.

This deepens your ability to work with these number types confidently and accurately.

algebraic expressions

Algebraic expressions made of complex numbers follow similar arithmetic rules to real numbers but incorporate additional steps due to the imaginary unit \( i \). The exercise demonstrates the fundamental process of expanding brackets in complex expressions.

After expanding, all terms are simplified by combining like ones—real with real, imaginary with imaginary,while correcting identity mistakes such as replacing \(-i^2\) with 2 (since \(i^2 = -1\)). Consider these as you handle such expressions:

• Expand naturally like polynomials, distributing each term.
• Apply correct identities, like \(i^2 = -1\).
• Simplify stepwise for clear, organized results aligned to standards.

This structured approach ensures consistency and correctness in calculations involving algebraic expressions of complex numbers.

error correction in mathematics

Error correction is a crucial part of learning mathematics, as mistakes often occur in complex problem solving. In the given exercise, not replacing \( i^2 = -1 \) was the key error. Recognizing and correcting mistakes helps in mastering complex numbers.

You can approach error correction through:

• Review each step systematically.
• Compare your approach to standard methods or solutions.
• Apply corrections to understanding why an error occurred and learn from it.

Mathematics isn't just about getting the right answer. It's about understanding the path to that answer. Spotting errors enhances comprehension and strengthens your foundational skills in topics like complex numbers.