Problem 27 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ 4 x^{2} y+5-6 x^{3} y-3 x^{2} y+2 x^{3} y $$

Step-by-Step Solution

Answer

The simplified polynomial is \(-4x^3y + x^2y + 5\).

Step 1: Identify Like Terms

In the given polynomial \(4x^2y + 5 - 6x^3y - 3x^2y + 2x^3y\), identify the like terms that can be combined. These are terms with the same variables raised to the same power. Here, \(4x^2y\) and \(-3x^2y\) are like terms, as well as \(-6x^3y\) and \(2x^3y\).

Step 2: Combine Like Terms

Combine the like terms identified in the previous step. For the \(x^3y\) terms, we have \(-6x^3y + 2x^3y = -4x^3y\). For the \(x^2y\) terms, \(4x^2y - 3x^2y = x^2y\). The constant term \(5\) has no like terms to combine with.

Step 3: Write the Polynomial in Descending Order

Now that the like terms have been combined, arrange the terms in descending order based on the power of \(x\). The descending order is: \(-4x^3y + x^2y + 5\).

Key Concepts

Understanding Like Terms in Polynomials

In algebra, especially when simplifying polynomials, recognizing "like terms" is crucial. Like terms in an expression are those that contain exactly the same variables raised to the same powers. Let's take the original polynomial from the exercise: 4x^2y + 5 - 6x^3y - 3x^2y + 2x^3y. Here:

• The terms \(4x^2y\) and \(-3x^2y\) are like terms because they both contain the same variables raised to the powers of 2 for \(x\) and 1 for \(y\).
• Similarly, \(-6x^3y\) and \(2x^3y\) are like terms, sharing the same variables raised to the power of 3 for \(x\) and 1 for \(y\).

Constant terms or standalone numbers, like \(5\) in our expression, are also considered as they are but are distinct and only combine with other constants. Recognizing which terms can be combined will simplify the process of further reducing and simplifying the polynomial.

Arranging in Descending Order

After identifying and combining like terms, organizing the polynomial in a specific order, particularly descending order, is key. Descending order simply means arranging the terms from the highest power of the variable to the lowest. For the polynomial given, we should first identify the highest power, which in this case is the cube term \(-4x^3y\). Placing terms in descending order makes them easier to read and understand, especially when comparing polynomials.For this example, we have:

• \(-4x^3y\): the highest power of \(x\) in the polynomial.
• \(x^2y\): the next highest power.
• \(5\): the constant, with no \(x\) or \(y\).

Ending with the constant term or any term without the core variable, this method creates a neat, coherent expression order that’s both consistent and preferred in mathematical writing.

Combining Like Terms for Simplification

Combining like terms is one of the fundamental steps in polynomial simplification. This process reduces the complexity of an equation, making it easier to solve or analyze. In our exercise, this step required us to combine like terms as follows:

• For terms like \(x^3y\): add \(-6x^3y\) and \(2x^3y\) to get \(-4x^3y\).
• For terms like \(x^2y\): add \(4x^2y\) and \(-3x^2y\) to yield \(x^2y\).

The lone number \(5\) is kept as it has no like terms. By focusing on achieving identical powers and variables, this manipulation is efficient and makes the polynomial easier to interpret and further manipulate if needed.