Problem 62 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -0.6(2 x+1)-0.1 $$

Step-by-Step Solution

Answer

The expression simplifies to \\(-1.2x - 0.7\\).

Step 1: Apply the Distributive Property

To eliminate the parentheses in the expression, we will use the distributive property. This means multiplying each term inside the parentheses by \(-0.6\). Thus, \(-0.6(2x + 1)\) becomes \(-0.6 \times 2x - 0.6 \times 1\).

Step 2: Distribute -0.6 Inside the Parentheses

Multiply \(-0.6\) by \(2x\) to get \(-1.2x\). Next, multiply \(-0.6\) by \(1\) to get \(-0.6\). The expression now is \(-1.2x - 0.6\).

Step 3: Combine Like Terms

Add the constant term from the previous step with \(-0.1\). Combine \(-0.6\) and \(-0.1\) to get \(-0.7\). Therefore, the simplified expression is \(-1.2x - 0.7\).

Key Concepts

Simplifying Expressions

When you come across algebraic expressions, the goal is often to make them as simple as possible. This is called simplifying expressions. You eliminate unnecessary parts, making the expression less cluttered and easier to work with. Here's how it works:

• Start by looking for parentheses. Use methods like the distributive property to remove them.
• Next, look for like terms that can be combined. Oh, don’t worry, we’ll cover that soon.
• Finally, write the expression in its simplest form — usually with terms ordered by degree in decreasing order.

Simplifying is like decluttering your room. Imagine finding what you need instantly because there’s less mess around you. The simpler the expression, the easier it is to use in further calculations.

Combining Like Terms

Once you have simplified as much as possible using the distributive property, the next step is to combine like terms. But what are like terms? They’re terms that have the same variable that is raised to the same power, even if they have different coefficients.

• For instance, in the expression \( -1.2x - 0.6 - 0.1 \), there are no "like terms" for the variable \( x \) but there are like terms among the constants, \( -0.6 \) and \( -0.1 \).
• Adding these together gives you \( -0.7 \), which simplifies your expression further.

Combining like terms is an essential skill because it helps to make expressions as straightforward as possible. It's like grouping similar items together in a grocery store aisle — everything becomes more organized!

Multiplying Polynomials

While our original exercise didn’t involve a full-fledged multiplication of polynomials, it's crucial to understand this concept. Multiplying polynomials means taking each term in one polynomial and multiplying it by each term in the other.

• If you have a monomial multiplied by a polynomial, distribute the monomial to each term inside the polynomial. That was exactly like our step when distributing \( -0.6 \) earlier.
• For polynomials multiplying other polynomials, use the FOIL method for binomials, or distribute each term accordingly.
• Take care to combine like terms after multiplication to simplify your expression as much as possible.

Practicing multiplication of polynomials with smaller expressions helps build the foundation for understanding more complex algebraic concepts. Think of it like practicing a musical instrument. You may start with simple notes, but soon you're playing a beautiful melody!