Problem 41 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ -4(x+2) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(-4(x+2)\). Answer: \(-4x - 8\)

Step 1: Distribute the multiplication

Multiply -4 by each term inside the parentheses. $$ -4(x+2) = -4 \cdot x + (-4 \cdot 2) $$

Step 2: Perform the multiplication

Now calculate the products: $$ -4 \cdot x + (-4 \cdot 2) = -4x - 8 $$

Step 3: Simplification

Since there are no like terms to combine, the expression is already simplified: $$ -4x - 8 $$ The final simplified expression is \(-4x - 8\).

Key Concepts

Distributive Property

In algebra, the distributive property is a useful tool for simplifying expressions and solving equations. It enables you to multiply a single term by each term within a set of parentheses.

For instance, if you come across an equation like \( -4(x+2) \), you apply the distributive property by multiplying \( -4 \) by \( x \) and \( 2 \) separately. The process goes as follows:

• First, multiply \( -4 \) by \( x \) to get \( -4x \).
• Then, multiply \( -4 \) by \( 2 \) to get \( -8 \).
• Thus, the expression becomes \( -4x - 8 \).

This method clears out the parenthesis and prepares you to further simplify the expression. Remember, the distributive property is vital whenever you encounter parentheses in algebraic expressions.

Combining Like Terms

To streamline an algebraic expression, it's essential to combine like terms. Like terms are terms that have the same variables raised to the same power.

In our example, after using the distributive property, we get \( -4x - 8 \). Although the expression is relatively simple, let's discuss what to do if it were more complex. If you had additional terms like \( +3x \) or \( -2 \), you would combine them with the matching terms in our existing expression.

Steps to Combine Like Terms:

• Add or subtract coefficients of identical variables. For instance, \( -4x \) and \( +3x \) would become \( -x \).
• Add or subtract constants. If another -2 were present, it would combine with -8 to make -10.

Remember to only combine terms with matching variable parts. This will often lead you to a more concise and manageable solution.

Algebraic Multiplication

Multiplication is one of the fundamental operations in algebra that involves combining coefficients and increasing variable powers when necessary. For a simple algebraic multiplication like \( -4 \) times \( x \) or \( 2 \) in the expression \( -4(x+2) \), you only deal with multiplying coefficients by variables and constants.

However, when multiplying two algebraic expressions, such as \( (a + b)(c + d) \), you distribute each term of the first expression over the second. It is crucial to multiply every term in the first expression with every term in the second one. It might seem complicated with more terms, but the principle remains the same: systematically multiply and then combine like terms as discussed above to simplify the expression fully.

Understanding how to perform algebraic multiplication accurately is critical for more advanced algebra and calculus concepts that you may encounter in your studies.