Problem 52 - Algebra 2

Question

Find the sum of \(i x^{2}-(2+3 i) x+2\) and \(4 x^{2}+(5+2 i) x-4 i\)

Step-by-Step Solution

Answer

The sum is \((4+i)x^2 + (3-i)x + 2-4i\).

Step 1: Write down the polynomial expressions

We are given two polynomials:1. \( i x^{2}-(2+3i)x+2 \) 2. \( 4x^{2}+(5+2i)x-4i \) Our goal is to find their sum.

Step 2: Combine like terms

Combine the coefficients of corresponding powers of \( x \) from both polynomials. - For \( x^{2} \) terms: \( ix^2 + 4x^2 = (i + 4)x^2 \)- For \( x \) terms: \(-(2+3i)x + (5+2i)x = ((5+2i) - (2+3i))x = (3-i)x \)- For constant terms: \( 2 - 4i \)

Step 3: Simplify each term

Simplify each group by performing the addition or subtraction:- \( (i + 4)x^2 \) remains as it is.- \( (3-i)x \) is already simplified.- \( 2 - 4i \) remains the same.

Step 4: Write the final polynomial

Combine all the simplified terms to express the sum of the polynomials:\[ (4+i)x^2 + (3-i)x + 2-4i \]

Key Concepts

Complex Numbers

Complex numbers are a fundamental part of algebra that students often encounter first in polynomial expressions. A complex number consists of a real part and an imaginary part. It's expressed in the form

• \( a + bi \)

where \( a \) is the real part, and \( bi \) is the imaginary part. The letter \( i \) is used to denote the imaginary unit, with the property that \( i^2 = -1 \).

When dealing with polynomial expressions involving complex numbers, as in our original exercise, you might encounter terms like \( (2+3i) \) or \( 4i \). These are handled using standard arithmetic rules – add or subtract the real and imaginary parts separately. For instance, with the term \((5+2i) - (2+3i)\), you subtract the real parts \((5-2)\) and the imaginary parts separately \((2i-3i)\), resulting in \(3-i\).

Working with complex numbers in polynomial addition involves similar principles as real numbers, but it also embraces the mysterious depth of imaginary numbers. To effectively simplify and handle these numbers, ensure to always pair and combine the real and imaginary components independently.

Polynomial Expressions

Polynomial expressions are algebraic expressions composed of variables raised to whole-number exponents along with coefficients. These expressions can have multiple terms, which are each a product of a coefficient and a variable term like \( x^2 \) or \( x \). In our exercise, the polynomials are expressed in terms involving complex numbers, such as

• \( ix^2 - (2+3i)x + 2 \)
• \( 4x^2 + (5+2i)x - 4i \)

To add polynomial expressions, line up terms with the same degree, or power of the variable. For instance, \( x^2 \), \( x \), and constant terms should be aligned. In the provided exercise, you align

• \( ix^2 \) with \( 4x^2 \),
• \(-(2+3i)x \) with \( (5+2i)x \), and
• the constants \( 2 \) with \(-4i \)

This pairing allows for straightforward addition or subtraction, as required.

Polynomial expressions are versatile and arise frequently across various fields of mathematics, making them worthy of mastering for any student.

Combining Like Terms

Combining like terms is a critical skill in algebra that simplifies expressions and equations. "Like terms" are terms that have the same variable raised to the same power. For instance, \( ix^2 \) and \( 4x^2 \) are like terms because both have the variable \( x^2 \). Similarly, terms like \( (5+2i)x \) and \( -(2+3i)x \) are considered like because they both have the variable \( x \) raised to the first power.

In the original problem, we simplify expressions by combining these like terms:

• For \( x^2 \), you calculate \( ix^2 + 4x^2 = (i+4)x^2 \) by simply adding the coefficients \( i \) and \( 4 \).
• For \( x \), you operate similarly, \( (5+2i)x - (2+3i)x = (3-i)x \), by subtracting the coefficients.
• Finally, the constants 2 and \(-4i\) stand alone as they are non-variable terms, combined as simply \( 2 - 4i \).

This method streamlines polynomial expressions and paves the way for solving more complex algebraic equations. Always be mindful to distinctly combine like terms only for an accurate simplified expression.