Problem 39 - Algebra 1: Concepts and Skills

Question

Use a horizontal format to add or subtract. $$ \left(x^{2}-7\right)+\left(2 x^{2}+2\right) $$

Step-by-Step Solution

Answer

The sum of the two given expressions is \(3x^{2}-5\).

Step 1: Identify like terms

In the sum, \((x^{2}-7)+(2x^{2}+2)\), the like terms are \(x^{2}\) and constant terms.

Step 2: Add the coefficients of the like terms

Add the coefficients of \(x^{2}\) which are 1 from \(x^{2}\) and 2 from \(2x^{2}\) to get \(3x^{2}\). Add the coefficients of the constant terms -7 and 2 to get -5.

Step 3: Write the sum

Combine the results from Step 2 to write the sum of the two expressions, \(3x^{2}-5\).

Key Concepts

Like Terms in Polynomials

When dealing with polynomials, a crucial concept is understanding "like terms." These are terms that contain the same variable raised to the same power, regardless of their coefficients. For instance, in the polynomial \[ax^n + bx^n + cx^m\], \(ax^n\) and \(bx^n\) are like terms because they both include the variable \(x\) raised to the power of \(n\).

• Like terms can be combined via addition or subtraction.
• They simplify polynomials during operations.

In our example, the like terms are \(x^2\) from \(x^2 - 7\) and \(2x^2\) from \(2x^2 + 2\). Recognizing like terms makes addition or subtraction straightforward, allowing us to simplify expressions effectively.

Understanding Coefficients

Coefficients are the numerical components in terms of polynomials. They appear in front of variables, indicating how many times that term is included in the polynomial. For instance, in \(3x^2\), \(3\) is the coefficient, suggesting that the term \(x^2\) is summed three times. Understanding coefficients is essential for:

• Addition: Combine coefficients of like terms.
• Checking polynomial equality.

In the original problem, we identified \(x^2\) with coefficients \(1\) from \(x^2 - 7\) and \(2\) from \(2x^2 + 2\). We then add \(1 + 2 = 3\) to obtain the new coefficient for \(x^2\), \(3x^2\). Similarly, the constant term coefficients \(-7\) and \(2\) were combined to give \(-5\).

Combining Expressions

An expression in algebra is a mathematical phrase that can involve numbers, variables, and operators (like addition and subtraction). Unlike equations, expressions do not have equal signs. Understanding how to manipulate expressions is essential for solving algebraic problems. When combining expressions:

• Ensure to maintain the integrity of each term.
• Operate only on like terms.
• Reorganize terms for clarity.

In this exercise, the goal was to combine \((x^2 - 7) + (2x^2 + 2)\) into a single expression. By identifying and summing up the like terms, the final result was simplified to \(3x^2 - 5\). This process illustrates how expressions can be transformed efficiently through simplification.