Problem 47 - Algebra A Combined Function

Question

Perform each indicated operation. $$ -5 x^{2}+x^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-4x^2\).

Step 1: Identify Like Terms

In the expression \[-5x^2 + x^2\]the terms are \(-5x^2\) and \(x^2\). Both terms contain \(x^2\), which means they are like terms and can be combined.

Step 2: Combine the Coefficients of Like Terms

To simplify, add the coefficients of the like terms:\[-5x^2 + 1x^2 = (-5 + 1)x^2 = -4x^2\]We add \(-5\) and \(1\), which gives \(-4\).

Step 3: Write the Simplified Expression

The simplified form of the expression is \[-4x^2\].This means we have combined the like terms into a single term with a coefficient of \(-4\).

Key Concepts

Polynomial Expressions

A polynomial expression is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's a series of terms, each containing both numbers and variables raised to whole number powers. For example, the expression \(-5x^2 + x^2\) is a polynomial expression featuring the variable \(x\) raised to the second power.

Polynomial expressions can be straightforward like \(x + 2\), or more complex, such as \(3x^3 - 2x^2 + x - 7\).

Understanding these expressions is key because they form the foundation of algebra. Being comfortable with polynomial expressions allows you to perform operations such as addition, subtraction, and further algebraic simplification.When dealing with polynomial expressions, always pay attention to:

• The degree of the expression, which is the highest power of the variable.
• The terms, which are parts of the expression separated by "+" or "-" signs.
• The coefficients, which are numbers multiplying the variables.

Recognizing these components can help in simplifying and manipulating polynomial expressions effectively.

Coefficients

Coefficients are the numbers that multiply the variables or powers of variables in a term. In the expression \(-5x^2 + x^2\), the coefficient of \(-5x^2\) is \(-5\), and for \(x^2\), it is \(1\). Coefficients indicate how many times the term is counted.

Recognizing coefficients helps in combining like terms, which is essential for simplifying expressions.

When combining terms, only terms with the same variable raised to the same power can be added or subtracted, per the coefficients that accompany them:

• Like terms have identical variable factors, such as \(x^2\).
• To combine like terms, add or subtract their coefficients. For \(-5x^2 + 1x^2\), this becomes \((-5 + 1)x^2 = -4x^2\).

Understanding coefficients is crucial for algebraic control and simplifies the process of handling complex polynomial expressions.

Algebraic Simplification

Algebraic simplification is the process of making expressions easier to work with by reducing complexity. This often involves combining like terms, and ensuring expressions are as concise as possible.

Combining like terms is a fundamental part of simplification. In the expression \(-5x^2 + x^2\), the like terms \(-5x^2\) and \(x^2\) were simplified by adding their coefficients to get \(-4x^2\). This creates a cleaner and simpler expression without changing its value.

Simple steps to simplify algebraic expressions include:

• Identify like terms that share the same variables and powers.
• Add or subtract the coefficients of these like terms.
• Rewriting the expression in its simplest form.

Simplification helps in better understanding of mathematical relationships and facilitates further calculations. Embracing simplification enhances problem-solving skills and allows complex problems to be tackled more efficiently.