Problem 57 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(2 x-3 x^{2}-x\) $$ (A)3 x-3 x^{2} $$ $$ (B)-2 x^{2} $$ $$ (C)x-3 x^{2} $$ $$ (D) 0 $$

Step-by-Step Solution

Answer

(C)x-3 x^{2}

Step 1: Identify Like Terms

Firstly, identify terms that are alike in this expression: \(2x\), \(-3x^{2}\), and \(-x\). In this case, \(2x\) and \(-x\) are like terms, while the term \(-3x^{2}\) stands alone since there are no other terms to the power of 2 within the expression.

Step 2: Combine Like Terms

Combine the like terms and simplify the polynomial. The like terms in this polynomial are \(2x\) and \(-x\) . Adding them together produces \(x\). Therefore, after combining like terms, the expression simplifies to \(x - 3x^{2}\).

Step 3: Final Expression

After simplifying, the final expression after combining the like terms is \(x - 3x^{2}\). This is the most simplified form of the given polynomial. This matches answer choice (C).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is fundamental when simplifying algebraic expressions. Like terms are terms within an expression that have the same variable raised to the same power.

For example, consider the algebraic expression from our exercise: \(2x - 3x^2 - x\). Here, \(2x\) and \(-x\) are like terms because they both contain the variable \(x\) raised to the first power. To simplify, we combine these by adding their coefficients. The coefficients of \(2x\) and \(-x\) are 2 and -1, respectively.

• \(2x + (-x) = 2x - x = x\)

After this combination, we are left with \(x - 3x^2\), which shows that the like terms have been effectively combined.

Common Pitfalls in Combining Like Terms

It's important to note that only coefficients can be combined, not the exponents. Students often mix different powers of a variable, which is incorrect. Only terms with exactly the same variable and exponent can be added or subtracted.

Polynomial Simplification

Polynomial simplification is a process of reducing expressions to their simplest form by performing operations like addition, subtraction, and, when possible, factoring.

A polynomial may look daunting at first, but the key is to organize and simplify step by step. With the given expression \(2x - 3x^2 - x\):

• We identified like terms.
• We combined them to reduce the polynomial to simpler terms.
• The expression then reached its simplest form \(x - 3x^2\).

This process ensures that the expression is as straightforward as possible, which is particularly helpful when using polynomials in further calculations or equations.

The Role of Coefficients

Coefficients are vital in polynomial simplification. Remember, when combining like terms, you are effectively combining their coefficients. The variable part, including the exponent, remains unchanged. Ensure to keep sign integrity while dealing with subtraction and addition to avoid errors.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables—such as \(x\) and \(y\)—and operators such as addition, subtraction, multiplication, and division. Simplifying these expressions is a significant part of algebra. This can involve expanding brackets, factoring, or other operations, with the goal of making the expression easier to use or understand.

In our exercise, \(2x - 3x^2 - x\) is an algebraic expression composed of terms that we combined appropriately. The art of simplifying expressions relies on understanding the properties of numbers and operations, as well as knowing how to spot and combine like terms.

Expression Versus Equation

It's also essential to distinguish an algebraic expression from an equation. An expression doesn't have an equality sign and doesn't state that two things are equal. Instead, it's something that can be simplified or evaluated for different values of its variables—much like a recipe that can produce different outcomes depending on the ingredients (variables) used.