Question

Use the Distributive Property to remove the parentheses. $$ (x-4)(x-2) $$

Short answer

The simplified expression is \( x^2 - 6x + 8 \).

Step-by-step solution

Apply the Distributive Property

Use the distributive property to expand the expression given by \( (x-4)(x-2) \). The distributive property states that \( a(b+c) = ab + ac \).

Distribute Each Term

Distribute each term in the first parenthesis to each term in the second parenthesis. This means: \( (x-4)(x-2) = x(x-2) + (-4)(x-2) \).

Perform the Distributions

Calculate each distribution: \( x(x-2) = x^2 - 2x \) and \( (-4)(x-2) = -4x + 8 \).

Combine Like Terms

Combine the like terms obtained from the previous step: \( x^2 - 2x - 4x + 8 \).

Simplify the Expression

Simplify the expression by adding the like terms: \( x^2 - 6x + 8 \).

Key concepts

Expanding Expressions

When we talk about expanding expressions, we refer to the process of removing parentheses by distributing each term inside the parentheses. This is often done using the distributive property. The distributive property states that for any numbers or variables, we have \( a(b + c) = ab + ac \).
In practice, if you have an expression such as \((x-4)(x-2)\), think of \(x-4\) and \(x-2\) as two entities that need to be distributed with each other. This means you will multiply every term in the first parenthesis by every term in the second parenthesis. Let’s apply this to our example:

• First, we distribute \(x\): \( x(x-2) \).
• Second, we distribute \(-4\): \( -4(x-2) \).

Combining these steps together, we have: \((x-4)(x-2) = x(x-2) + (-4)(x-2)\). Now we can calculate each part.

Combining Like Terms

Combining like terms refers to the process of merging terms that have the same variable raised to the same power. This makes expressions simpler and easier to understand. Revisiting our expanded form \(x(x-2) + (-4)(x-2)\):

• Calculate \(x(x-2)\): \( x^2 - 2x \).
• Calculate \((-4)(x-2)\): \( -4x + 8 \).

Now, we get \(x^2 - 2x - 4x + 8\).
The next step is to combine like terms. In this expression, the like terms are \(-2x\) and \(-4x\) because they both have the \(x\) variable. Adding these together, we get \( -2x - 4x = -6x \).
So, now our expression looks like \( x^2 - 6x + 8 \). Combining like terms simplifies the expression and makes it easier to work with.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest form. This often includes expanding expressions and combining like terms, as we’ve discussed. Our goal is to make the expression as concise as possible. Starting from the expression \( (x-4)(x-2) \), we expanded it to: \( x^2 - 2x - 4x + 8 \)
Combining like terms, we got: \( x^2 - 6x + 8 \).
Here, \( x^2 \), \(-6x\), and \( 8 \) are all distinct, and there are no further like terms to combine. This is the simplest form of this expression.

• We started with \((x-4)(x-2)\).
• We expanded it using the distributive property.
• We combined like terms to simplify it.

This process of expanding, combining, and simplifying is crucial in algebra, making complex expressions easier to understand and work with.