Problem 59 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 7 x(2-x)-4 x $$

Step-by-Step Solution

Answer

The simplified expression is \(10 x - 7 x^2\).

Step 1: Distribute the multiplication

Distribute the multiplication of \(7 x\) into the parenthesis: \(7 x \times 2 - 7 x \times x = 14 x - 7 x^2\). Now the expression becomes \(14 x - 7 x^2 - 4 x\).

Step 2: Combine like terms

Combine the \(14 x\) and \(-4 x\) terms to simplify the expression: \(14 x - 4 x = 10 x\). Then the expression becomes \(10 x - 7 x^2\).

Key Concepts

Simplification

When simplifying algebraic expressions, the goal is to make the expression as easy to work with as possible. Simplification involves reducing complexities by following basic algebraic rules. We aim to tidy up the expression by removing unnecessary parentheses and combining terms where possible.

In our exercise, we started with the expression:

• \[7x(2 - x) - 4x\]

The process involves systematically addressing each part of the expression to end up with something cleaner and easier to understand. After performing all steps accurately, your final expression should look neater and simpler, ready for any further operations or analyses you might need to perform.

Like Terms

The concept of like terms is central to simplifying expressions. Like terms are terms whose variables (and their exponents) are the same. This similarity allows us to combine them into a single term by simply adding or subtracting their coefficients.

For example, in the expression:

• \[14x\] and \[-4x\]

These terms are considered like terms because they both include the variable \(x\) raised to the first power. We can combine like terms by adjusting just the coefficients.

• \[14x - 4x = 10x\]

By understanding and recognizing like terms, we easily reduce the complexity of the expression and move one step closer to finding a simpler form.

Distribution

Distribution is a crucial technique in algebra that involves multiplying a term outside of a parenthesis by every term inside the parenthesis. This helps in removing the brackets, thus simplifying part of the expression.

In the example expression:

• \[7x(2 - x)\]

We distribute the \(7x\) across each term inside the parentheses:

• \[7x imes 2 = 14x\]
• \[7x imes -x = -7x^2\]

After distribution, the expression transforms from one involving parentheses to:

• \[14x - 7x^2 - 4x\]

It is critical to carefully apply distribution as a mistake here can affect the entire simplification process. By thorough distribution, we ensure every part of our expression is considered and correctly simplified.