Problem 12 - Algebra 1: Concepts and Skills

Question

Find the sum or the difference of the polynomials. $$ (7 x-3)-(9 x-2) $$

Step-by-Step Solution

Answer

The difference of the two polynomials is \(-2x - 1\).

Step 1: Distribute the Negative

In the first step, distribute the negative sign to every term inside the parenthesis. Therefore, the equation now becomes \(7x - 3 - 9x + 2 \).

Step 2: Group like Terms

Next, group like terms together. This results into two categories: \(7x - 9x\) for the x terms, and \(-3 + 2\) for the constant terms.

Step 3: Subtract like Terms

Subtract the common terms accordingly to arrive at the solution: x terms \(7x - 9x = -2x\) and the constant terms \(-3 + 2 = -1\) .

Step 4: Combine to form the Resulting Polynomial

Finally, combine the x terms and the constant term to form the resulting polynomial which is \(-2x - 1\)

Key Concepts

Distribute the Negative Sign

In polynomial subtraction, distributing the negative sign is an essential first step. When you see a negative sign in front of a set of parentheses, it means that the negative should be applied to each term within the parentheses. This action fundamentally changes the signs of those terms. For example, considering the original exercise, the expression \((7x - 3) - (9x - 2)\) involves distributing the negative sign from in front of the second set of parentheses:

• The term \(9x\) becomes \(-9x\).
• The term \(-2\) becomes \(+2\).

This results in simplifying the expression to \(7x - 3 - 9x + 2\). Distributing the negative correctly sets the stage for effectively simplifying the problem later on.

Like Terms

Understanding like terms is critical when working with polynomials.Like terms are terms within an expression that contain the same variable raised to the same power. In the simplified polynomial subtraction from our example, the like terms are:

• \(7x\) and \(-9x\) because they both have the variable \(x\).
• Constant terms \(-3\) and \(+2\), as they lack variables and are therefore grouped together as like terms.

Grouping like terms simplifies the process of combining them to further simplify the polynomial.Properly identifying and reorganizing like terms is crucial for the next step: actually subtracting them.

Subtract Like Terms

Once like terms are grouped, the process of subtracting them becomes straightforward. Subtracting like terms means directly performing arithmetic operations on the coefficients of similar terms.Let's look at our example closely:

• For the x terms: \(7x - 9x = -2x\). You simply subtract 9 from 7, which results in \(-2\), keeping the variable \(x\) consistent.
• For the constant terms: \(-3 + 2 = -1\). You add 2 to \(-3\), a simpler arithmetic operation without any variables involved.

When all like terms have been subtracted or added appropriately, they can be recombined to give you the final simplified polynomial.In our case, you end up with the polynomial \(-2x - 1\). This methodical approach ensures accuracy in obtaining the correct polynomial form.