Problem 10 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$6 y+4 y^{2}-6 y+y^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(5y^2\).

Step 1: Identify Like Terms

Observe the given expression: \(6y + 4y^2 - 6y + y^2\).Like terms are terms whose variables and their exponents match. Here, the like terms are: - \(6y\) and \(-6y\) (they both have the variable \(y\)),- \(4y^2\) and \(y^2\) (they both have the variable \(y^2\)).

Step 2: Combine Like Terms for \(y\)

Combine the like terms involving \(y\):\(6y - 6y = 0\).These terms cancel each other out, so they disappear from the expression.

Step 3: Combine Like Terms for \(y^2\)

Combine the like terms involving \(y^2\):\(4y^2 + y^2 = 5y^2\).Add the coefficients (4 and 1) to get the result of \(5y^2\).

Step 4: Write the Simplified Expression

The expression now only includes the term \(5y^2\) as all other terms have been combined or canceled out.

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, constructed using operations like addition, subtraction, and multiplication. The terms in a polynomial are typically written in descending order of their degree, which is the highest power of the variable present. For example, in the expression \(4y^2 + 6y - 6y + y^2\), each part like \(4y^2\) or \(6y\) is called a term. Here:

• \(4y^2\) and \(y^2\) are terms with \(y^2\) as their highest power.
• \(6y\) and \(-6y\) are first-degree terms.

Polynomials can have many terms, but the ones addressed here have like terms that can be simplified, making calculations easier and more manageable.

Algebraic Expressions

At the heart of algebra are algebraic expressions, combinations of numbers, variables, and operations. They are akin to phrases in mathematics that help us solve equations and model real-life situations. In the expression \(6y + 4y^2 - 6y + y^2\), each piece is a distinct term that's part of the whole expression. These terms are:

• \(6y\) and \(-6y\), which are linear terms with the variable \(y\).
• \(4y^2\) and \(y^2\), which are quadratic terms because of the \(y^2\).

When working with algebraic expressions, it's crucial to identify like terms, which can be combined to simplify equations and make calculating more straightforward.

Simplification of Expressions

Simplifying expressions is a fundamental skill in algebra that involves combining like terms to make expressions easier to handle. For the given expression \(6y + 4y^2 - 6y + y^2\), simplifying includes steps like:

• Identifying like terms: Terms that have identical variable parts. For example, \(6y\) and \(-6y\) are like terms because both contain the variable \(y\).
• Combining like terms: Adding or subtracting their coefficients. Here, \(6y - 6y\) results in \(0\), effectively eliminating these terms.
• Adding the coefficients of terms like \(4y^2\) and \(y^2\), which results in \(5y^2\).

The final simplified expression, \(5y^2\), is cleaner and simpler, highlighting the power of combining like terms in mathematics.