Problem 7 - Elementary Algebra

Question

Simplify each of the following expressions by using the distributive property and combining like terms. $$ 4(x+6)+3\left(2+x+3 x^{2}\right)-2 x^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression using the distributive property and combining like terms: $$4(x+6)+3(2+x+3x^2)-2x^2$$ Answer: The simplified expression is $$7x^2 + 7x + 30$$.

Step 1: Distribute each term to the expressions inside the parentheses

To do this, multiply each term outside of the parentheses with each term inside the brackets, as follows: $$ 4(x+6)+3(2+x+3x^2)-2x^2 $$ $$ = 4x + 24 + 6 + 3x + 9x^2 - 2x^2 $$

Step 2: Combine like terms

Now, group the terms with the same variables and powers, and add or subtract them as necessary: $$ = (4x + 3x) + (24 + 6) + (9x^2 - 2x^2) $$

Step 3: Simplify the expression

Finally, add or subtract the terms in each group to simplify the expression: $$ = 7x + 30 + 7x^2 $$ So, the simplified expression is: $$ 7x^2 + 7x + 30 $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions by distributing a multiplier to each term within parentheses.

Imagine you are handing out apples to potential groups of people, and you've got to ensure everyone gets the same number of apples. The distributive property does the same with numbers and variables, ensuring each term inside the parentheses is multiplied by the term outside.

For example, if we have a coefficient like 3 outside the parentheses \(3(x + 2)\), we multiply both \(x\) and \(2\) by \(3\), resulting in \(3x + 6\). It's a way to distribute the multiplication evenly across all terms within the brackets.

This property is essential when you encounter expressions like \(4(x+6)\), where you'll apply the property to multiply \(4\) times \(x\) and \(4\) times \(6\), resulting in the expression \(4x+24\).

Combining Like Terms

After using the distributive property, your next step is often to 'combine like terms.' This term simply means to add or subtract terms that have the same variable and the same exponent.

In our social example, think about combining like terms as grouping friends who wear the same color shirt. Each group represents terms with a similar factor—a shared identifying component (the variable and exponent, or the color of the shirt).

For instance, with \(4x + 3x\), both terms have the variable \(x\) to the power of 1, so they can be added together to make \(7x\). It helps to visualize terms as joining together to form a more straightforward, compact group. Remember, constants (numbers without variables, like \(24 + 6\)) are also like terms and can be combined in the same way, resulting in \(30\).

In the context of our expression, after distributing, we combine terms with the same power of \(x\) to reach a simpler form.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to a form that is more useful or easier to understand. It's like solving a puzzle, where your goal is to move and transform the pieces until you can clearly see the big picture.

This process often involves several steps, including the use of distributive property and combining like terms, as we've seen in this exercise. But it also involves other techniques, such as factoring, expanding, and applying exponent rules.

In our example, we've already distributed and combined like terms. Next, we ensure our equation takes the best shape by adding or subtracting coefficients and making sure the terms are in descending order of power for \(x\), which is a typical convention in algebra. Thus, our final simplified form \(7x^2 + 7x + 30\) showcases this. By gaining competence in algebraic manipulation, you become adept at transforming complex algebraic expressions into their most workable and understandable forms.