Problem 16 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ 17 y^{2}-22 y-y^{2} $$

Step-by-Step Solution

Answer

The simplified polynomial is \(16y^2 - 22y\).

Step 1: Identify Like Terms

Begin by identifying the like terms in the polynomial. The terms are \(17y^2\), \(-22y\), and \(-y^2\). Notice that \(17y^2\) and \(-y^2\) are like terms because they both involve \(y^2\).

Step 2: Combine Like Terms

Next, combine the like terms \(17y^2\) and \(-y^2\). This is done by adding or subtracting their coefficients: \[ 17y^2 - y^2 = (17 - 1)y^2 = 16y^2 \]The term \(-22y\) does not have any like terms to combine with, so it remains unchanged.

Step 3: Write in Descending Order

Finally, arrange the polynomial in descending order of the power of \(y\). Since the highest power of \(y\) is in \(16y^2\), and the next term is \(-22y\), the polynomial is: \[ 16y^2 - 22y \]

Key Concepts

Like Terms

When dealing with polynomials, understanding what constitutes 'like terms' is crucial. Like terms are those that have the same variables raised to the same powers. For instance, in the polynomial \(17y^2 - 22y - y^2\), the terms \(17y^2\) and \(-y^2\) are like because they both include \(y^2\). Identifying like terms allows you to simplify expressions by combining them, which makes working with polynomials easier. Here are some tips to help you spot like terms:

• Look for terms with the same variable and exponent, such as \(x^2\) and \(2x^2\).
• Remember that numbers (constants) are like terms too, e.g., \(5\) and \(-3\) can be combined.
• Terms must precisely match in variables and powers to be combined; \(xy\) and \(yx\) are like terms, but \(xy\) and \(x^2\) are not.

This principle is foundational in polynomial manipulation, simplifying expressions, and solving equations. Always start by identifying the like terms before any simplification.

Combining Like Terms

Once you’ve identified like terms in your polynomial, the next step is to combine them to simplify the expression.This process involves adding or subtracting their coefficients while keeping the variable parts unchanged. In the given polynomial \(17y^2 - 22y - y^2\), we combine the coefficients of the like terms \(17y^2\) and \(-y^2\):

• Calculate the new coefficient: \(17 - 1 = 16\)
• Therefore, \(17y^2 - y^2 = 16y^2\)

The term \(-22y\) has no counterparts, so it stays as is. Always ensure to perform operations only on coefficients while keeping the variable and exponent unchanged.Simplifying polynomials through combining like terms results in a neater and often more manageable expression, which can be vital for solving polynomial equations and analyzing their behaviors.

Descend Order Polynomials

Polynomials should ideally be expressed in descending order by the power of the variable.This order starts with the highest exponent term and finishes with the lowest. Given the simplified polynomial \(16y^2 - 22y\), let's put it in proper descending order:

• Identify which term has the highest power of \(y\).
• \(16y^2\) comes first as it involves \(y^2\).
• \(-22y\) comes next since it’s \(y^1\) or simply \(y\).

Writing polynomials in descending order is commonly required for standardized formats, making it easier to read and work with, especially when performing operations such as addition, subtraction, or solving equations.This particular order highlights the dominant power and influences how the polynomial behaves as a mathematical function.