Problem 11 - Algebra 1: Concepts and Skills

Question

Find the sum or the difference of the polynomials. $$ (2 x-9)+(x-7) $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(3x - 16\).

Step 1: Identify Like Terms

In the given polynomial expressions, the like terms are identified as those which contain the variable \(x\) and those which are constants. So, in this case, \(2x\) and \(x\) are like terms, and \(-9\) and \(-7\) are like terms.

Step 2: Add the Like Terms

Add the coefficients of the like terms together to get the new coefficients. The coefficient of \(x\) in the first expression is 2 and in the second expression is 1. Adding these together gives \(2+1=3\), so the sum of the \(x\) terms is \(3x\). Similarly, adding the constant terms together gives \(-9 - 7 = -16\).

Step 3: Solution Expression

The final solution is obtained by writing down all the sums of the like terms. Here there are two: one for the \(x\) terms and one for the constant terms. Therefore, the final solution is \(3x - 16\).

Key Concepts

Like Terms

When we work with polynomials, it's crucial to recognize 'like terms'. Like terms are terms that contain the same variables raised to the same power. This means they look very similar, but the coefficients can differ. For example, in the polynomial expression \( (2x - 9) + (x - 7) \), \( 2x \) and \( x \) are like terms because they both have the variable \( x \) raised to the power of 1. Similarly, \( -9 \) and \( -7 \) are constants, which are also considered like terms. Identifying and combining like terms is the first step in simplifying algebraic expressions.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomials. In this exercise, we are focusing on addition and subtraction.
To add or subtract polynomials, follow these simple steps:

• Identify the like terms in each polynomial.
• Add or subtract the coefficients of the like terms.
• Write the result by combining the new coefficients with their common variable.

For our example, adding \( 2x \) and \( x \) gives \( 3x \), while subtracting \( 9 \) from \( -7 \) gives \( -16 \). Combining these, we arrive at the final result: \( 3x - 16 \). Understanding these operations helps in further complex calculations involving algebraic expressions.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation (like addition or subtraction). Unlike equations, algebraic expressions do not contain an equal sign. They can represent real values where variables can have multiple possibilities.
It's important to master how to handle them to solve algebra problems efficiently. Let's break down the algebraic expression from our exercise:

• The expression \( (2x - 9) + (x - 7) \) involves two separate polynomials.
• Typically, polynomials are expressions made up of multiple terms connected by addition or subtraction.
• The aim is to simplify them by combining like terms, as we did earlier.

Recognizing how these expressions are structured and the purpose each part serves aids in systematically approaching more complex algebraic tasks.