Problem 59 - Algebra 1: Concepts and Skills

Question

Simplify the expression by combining like terms if possible. If not possible, write already simplified. $$ 3 w^{2}+2 w-3 w $$

Step-by-Step Solution

Answer

The simplified expression is \(3w^{2} - w\).

Step 1: Identify and Combine like terms

Look for terms that have the same variable and exponent. Here, \(2w\) and \(-3w\) are like terms. Combining these two together, we compute \(2w - 3w\), which gives \(-w\)

Step 2: Write the final, simplified expression

Lastly, write down the final expression. The like terms will be replaced by their combination, and any other terms will remain intact. So, \(3w^{2}\) stays as it is and \( 2w - 3w\) becomes \(-w\). The final expression is thus \(3w^{2} - w\).

Key Concepts

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions. It's about grouping terms with the same variable raised to the same power, making the expression easier to work with. In the exercise, you are presented with the expression \(3w^{2} + 2w - 3w\).
To combine like terms, follow these steps:

• Identify terms with the same variable and exponent. Here, \(2w\) and \(-3w\) are alike because they both have the variable \(w\) and no exponents other than 1.
• Add or subtract their coefficients. The coefficients are 2 and -3. Calculating \(2 - 3\), we get \(-1\). Thus, \(2w - 3w = -w\).

Once combined, the terms much like simplification, allow for a clearer view of the expression's structure, which can be crucial for solving more complex problems.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operational signs. They model real-life situations and give us a flexible language to express and solve equations.
Here are some basic components:

• **Variables**: Symbols (like \(w\), \(x\), or \(y\)) that represent unknown values.
• **Coefficients**: Numbers in front of variables, indicating multiplication. In \(3w^2 + 2w - 3w\), 3, 2, and -3 are coefficients.
• **Constants**: Stand-alone numbers with no variables, though not in this exercise.
• **Operators**: Signals like '+' or '-', dictating how terms interact.

The expression \(3w^2 + 2w - 3w\) combines these elements, demonstrating how algebra can compactly convey mathematical ideas.

Polynomials

Polynomials are special types of algebraic expressions. They consist of many terms, either constants, or products of constants and variables raised to whole-number exponents.
They are categorized based on degrees and the number of terms:

• **Degree**: Highest exponent of a variable. Here, it's \(w^2\), making this a quadratic polynomial.
• **Mono-, Bi-, and Trinomials**: Descriptive of the number of terms, though such specific classification isn't emphasized here.

Polynomials are crucial in algebra for modeling and analysis. They allow huge flexibility in problem-solving, particularly when graphed as curves or surfaces. In the expression \(3w^2 + 2w - 3w\), simplifying using like terms delivers a streamlined, quadratic polynomial: \(3w^2 - w\). Understanding their structure helps build foundational skills for further algebraic study.