Problem 33 - Algebra 1

Question

USING THE DISTRIBUTIVE PROPERTY Use the distributive property to simplify the expression. $$(x+3) x$$

Step-by-Step Solution

Answer

The simplified expression for \((x+3) x\) is \(x^2 + 3x\).

Step 1: Distributive Property

The distributive property states that for all real numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. We apply this to \((x+3) x\), where \(a=x\), \(b=x\), and \(c=3\).

Step 2: Apply the distributive property

We rearrange \((x+3) x\) as \(x*x + x*3\).

Step 3: Simplify the Expression

The result simplifies to become \(x^2 + 3x\).

Key Concepts

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions involves reducing them to their simplest form, which makes them easier to understand and work with. This typically involves combining like terms, which are terms that have the same variable raised to the same power, and performing basic arithmetic operations.

For example, consider the expression \(2x + 3x - x\). To simplify this, we combine the like terms which all contain the variable \(x\). Adding up the coefficients (2, 3, and -1) gives us \(4x\). The expression \(2x + 3x - x\) simplifies to \(4x\), which is a much cleaner and clearer form.

Simplifying expressions is crucial in solving equations and performing operations such as addition, subtraction, and factoring polynomials. It's the first and very important step in tackling algebraic problems.

Applying the Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term in a parenthesis individually. It's often used to expand algebraic expressions and is a key tool when simplifying expressions and solving equations.

For instance, when we apply the distributive property to the expression \(a(b + c)\), we multiply \(a\) by both \(b\) and \(c\) separately and then add the products. This gives us \(ab + ac\), which is the expanded form of our initial expression.

The distributive property not only simplifies expressions but also helps to make connections between algebraic procedures and arithmetic operations. Recognizing when and how to apply the distributive property is essential for success in algebra.

Multiplying Polynomials

When it comes to multiplying polynomials, the operation extends the distributive property over multiple terms. Consider the multiplication of two binomials, \(p + q\) and \(r + s\). To multiply them, we need to apply the distributive property to each term in the first polynomial with every term in the second polynomial.

This process is often referred to as FOIL (First, Outside, Inside, Last), indicating the order in which we distribute the terms:

• First: Multiply the first terms in each polynomial.
• Outside: Multiply the outermost terms.
• Inside: Multiply the innermost terms.
• Last: Multiply the last terms in each polynomial.

After multiplying, we then combine like terms to simplify the expression to its final form. This process is the backbone for understanding more complex polynomial operations such as square and cube of binomials, factoring, and polynomial division.