Question

Use the Distributive Property to remove the parentheses. $$ (x-8)(x-2) $$

Short answer

Expand to: \[x^2 - 10x + 16\]

Step-by-step solution

Identify the Distributive Property

The Distributive Property states that for any numbers or variables, \[a(b + c) = ab + ac\]. In this case, we need to apply the property to remove the parentheses from the expression \[(x-8)(x-2)\].

Distribute Each Term

Use the Distributive Property to expand the expression: \[(x-8)(x-2) = x(x-2) - 8(x-2)\]

Multiply Each Term

Now simplify by distributing each term within the parentheses: \[x(x-2) - 8(x-2) = x \times x + x \times (-2) - 8 \times x - 8 \times (-2)\]This simplifies to: \[x^2 - 2x - 8x + 16\]

Combine Like Terms

Combine the like terms to further simplify the expression: \[x^2 - 10x + 16\]

Key concepts

Distributive Property

The Distributive Property is a fundamental rule in algebra that lets you simplify expressions. It states: \[a(b + c) = ab + ac\]. Using this property helps you remove parentheses in algebraic expressions, making them easier to work with. For example, in the exercise \[(x-8)(x-2)\], we apply the distributive property to break down the expression. Remember, when you see distributive, think about 'distributing' the multiplication over addition or subtraction inside the parentheses.

Expanding Expressions

Expanding expressions involves using the Distributive Property to 'spread out' a product over addition or subtraction. In our example, we start with \[(x-8)(x-2)\]. To expand this, distribute each term: \[x(x-2) - 8(x-2)\]. This step is crucial because it transforms a complicated expression into simpler, manageable parts. Notice how each term outside the parentheses is multiplied by every term inside the parentheses. This helps in breaking down complex expressions into simpler forms.

Combining Like Terms

After expanding expressions, the next important step is to combine like terms. Like terms are terms that contain the same variables raised to the same power. For our example, after expanding, we get: \[x^2 - 2x - 8x + 16\]. Here, \(-2x\) and \(-8x\) are like terms because they both contain the variable \(x\) to the first power. Combining these terms gives us \[-10x\]. This step simplifies the expression into fewer terms while keeping mathematical integrity.

Polynomial Simplification

Polynomial simplification is the process of making a polynomial easier to work with by combining like terms and arranging them in a standard form. In the final step of our solution, the expression \[x^2 - 2x - 8x + 16\] simplifies to \[x^2 - 10x + 16\]. This is the simplest form of the polynomial. Simplifying polynomials makes solving equations or further algebraic manipulations more straightforward. Always arrange terms in descending order of their exponent values to follow the standard polynomial form.