Problem 18 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 5 y-1210-y y 2 $$

Step-by-Step Solution

Answer

The simplified expression is \(-y^3 + 5y - 22\).

Step 1: Identify Terms

The given expression is a combination of terms involving variables and coefficients: \(5y - 12 - 10 - yy^2\). Here, "y" can be considered as \(y\) and "yy^2" stands for \(y \times y^2\), which is \(y^3\).

Step 2: Group Like Terms

Rewrite the expression \(5y - 12 - 10 - yy^2\) by grouping like terms together. We separate the terms involving the variable \(y\) from the constants and powers of \(y\). So, the expression becomes \(5y - y^3 - 12 - 10\).

Step 3: Simplify Constants

Combine the constant terms \(-12\) and \(-10\). This simplifies to \(-22\). So the expression becomes \(5y - y^3 - 22\).

Step 4: Arrange in Standard Form

Arrange the terms in descending order of the power of \(y\). The standard form for polynomials is usually written from the highest degree to the lowest. Therefore, rewrite \(5y - y^3 - 22\) as \(-y^3 + 5y - 22\).

Key Concepts

Like Terms

In algebra, the concept of "like terms" refers to terms in an expression that have the same variable raised to the same power. For instance, in the expression \(5y - 12 - 10 - yy^2\), terms like \(5y\) and \(- y^3\) are considered different because they involve different powers of \(y\). Recognizing like terms allows you to effectively combine them during simplification.

• Only terms with exactly identical variables and powers are like terms. For example, \(3x^2\) and \(2x^2\) are like terms, but \(3x^2\) and \(2x\) are not.
• You can perform operations such as addition and subtraction only on like terms. This is a crucial step when simplifying polynomial expressions.

When simplifying expressions, look out for like terms to make the expression clearer and more manageable. In our original problem, the grouping of like terms helped simplify the expression to a more workable form.

Standard Form

The term "standard form" in the context of polynomials refers to writing the polynomial in a structured manner. Specifically, this means arranging the terms from the highest degree to the lowest degree.

• The highest degree term in a polynomial has the variable raised to the highest power. It's the starting point for writing a polynomial in standard form.
• This arrangement helps to easily identify the leading term, which is crucial for understanding the polynomial's behavior as the variable becomes very large or small.

Using standard form is particularly useful in communicating mathematics clearly and ensuring uniformity when working with polynomials. In our solution, the expression \(5y - y^3 - 22\) was reorganized into \(-y^3 + 5y - 22\) to conform to the standard form guideline.

Descending Order

Descending order is a term used to describe how we arrange the terms of an expression. When dealing with polynomials, it means sorting the terms from the highest power to the lowest power.

• This is critical for solving, simplifying, and comparing polynomials.
• It ensures that polynomials have a clear, easy-to-read structure, which is helpful for further operations such as addition, multiplication, or finding roots.

In the given problem, arranging the terms in descending order required us to place \(-y^3\) first, followed by \(5y\), and then \(-22\). This orderly approach simplifies the process for any operations or evaluations that follow. By always using descending order, mathematicians ensure consistency and clarity across different problems and exercises.