Problem 18 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ -3 y^{2}-y-6 y^{2}+7 y $$

Step-by-Step Solution

Answer

The simplified polynomial is \(-9y^2 + 6y\).

Step 1: Identify Like Terms

First, we must identify the like terms in the polynomial expression. Like terms are terms that have the same variables raised to the same power. In the given expression \[-3y^2 - y - 6y^2 + 7y\]the like terms are the terms with \(y^2\) and the terms with \(y\).

Step 2: Combine Like Terms

Now, let's combine the like terms:1. Combine the \(y^2\) terms: \(-3y^2 - 6y^2 = -9y^2\).2. Combine the \(y\) terms: \(-y + 7y = 6y\).

Step 3: Write the Simplified Expression

After combining like terms, write the simplified polynomial in descending order of powers:The simplified expression is:\[-9y^2 + 6y\]

Key Concepts

Combining Like Terms

In the world of polynomials, combining like terms is crucial for simplification. Like terms are terms within a polynomial that have the same variable raised to the same power. This means that terms with matching variables and exponents can be added or subtracted together.
For example, in the expression \(-3y^2 - y - 6y^2 + 7y\), you can identify that:

• \(-3y^2\) and \(-6y^2\) are like terms, both having the variable \(y\) raised to the power of 2.
• \(-y\) and \(7y\) are like terms because they share the same variable \(y\) raised to the power of 1.

After identifying these, you can add or subtract the coefficients of these like terms to combine them. By doing this, you simplify the polynomial significantly, making it easier to interpret and work with. Remember, only the coefficients change during this process— the variables and their exponents stay the same.

Descending Order of Powers

Arranging terms in descending order of powers is an important step in simplifying polynomials. This means rewriting the polynomial so that the terms are ordered from the highest to the lowest power of the variable. This convention helps in maintaining a clear and organized expression.
For instance, the expression\(-3y^2 - y - 6y^2 + 7y\) first needs identifying the like terms. After combining the like terms to get \[-9y^2 + 6y\], it is crucial to arrange this result with the highest exponent term appearing first. The term \(-9y^2\) (where the exponent is 2) precedes \(6y\) (where the exponent is 1).
This technique not only aids in clarity but also prepares the polynomial for further operations, such as factoring or solving equations which often rely on such standardized forms.

Simplifying Polynomials

Simplifying polynomials is the process of taking a complex polynomial and making it as simple as possible. This involves several steps, crucial among them being combining like terms and arranging the terms in descending order of powers.
To simplify \(-3y^2 - y - 6y^2 + 7y\), you would:

• Identify and combine like terms: reducing it to \(-9y^2 + 6y\).
• Arrange these terms in descending order of their power, which correctly aligns the expression as \(-9y^2 + 6y\) already shows.

The goal of simplification is to have the expression in the simplest form possible.
This makes it easier to graph, integrate, differentiate, or handle in any additional mathematical procedures needed. Simplified polynomials are far easier to interpret and use, presenting essential information at a glance.