Problem 9 - Algebra 2

Question

Simplify. $$ (x+6)(x+3) $$

Step-by-Step Solution

Answer

The simplified form is \(x^2 + 9x + 18\).

Step 1: Apply the Distributive Property

To simplify \((x+6)(x+3)\), we apply the distributive property \((a+b)(c+d) = ac + ad + bc + bd\). Let's start by distributing each term in the first binomial across the second binomial.

Step 2: Distribute 'x' Across '(x+3)'

First, distribute the \(x\) from \((x+6)\) across \((x+3)\). Multiply each term inside the second bracket by \(x\):\[x(x+3) = x^2 + 3x\].

Step 3: Distribute '6' Across '(x+3)'

Next, distribute the \(6\) from \((x+6)\) across \((x+3)\). Perform the multiplications as follows:\[6(x+3) = 6x + 18\].

Step 4: Add Like Terms

Combine the results from steps 2 and 3:\[x^2 + 3x + 6x + 18\]. The like terms here are \(3x\) and \(6x\). Add them together to get:\[x^2 + 9x + 18\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms inside parentheses. It is especially useful when dealing with expressions like \( (x+6)(x+3) \) because it helps break down multiplication into simpler parts. The distributive property is expressed as: \\((a+b)(c+d) = ac + ad + bc + bd\). This means you multiply each term in the first set of parentheses by every term in the second set.

• Multiply the first term from the first binomial by each term in the second binomial.
• Repeat this process for each subsequent term in the first binomial.

In our exercise, we first multiply \( x \) in \( (x+6) \) by \( x \) and then by +3. Then, we do the same for 6 in \( (x+6) \). This way, every term in \( (x+6) \) is distributed over every term in \( (x+3) \), simplifying the multiplication process.

Combining Like Terms

After distributing, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, after distributing we find ourselves with the terms \( x^2 + 3x + 6x + 18 \).

• The \( x^2 \) is a standalone term, as it's the only one of its kind.
• The terms \( 3x \) and \( 6x \) are considered like terms because they both contain the variable \( x \) raised to the first power.

By adding the coefficients of the like terms \( 3x + 6x \), you get \( 9x \). This results in the simplified expression \( x^2 + 9x + 18 \). Always look out for like terms to combine after using the distributive property, as it can further simplify your expression and make it easier to interpret.

Binomials

Understanding binomials is key when learning about polynomial multiplication. A binomial is an algebraic expression that contains exactly two terms, often connected by a plus or minus sign. In our case, \( (x+6) \) and \( (x+3) \) are examples of binomials.

• Each binomial can be thought of as a simple polynomial with two distinct terms.
• When multiplying binomials, each term in the first binomial must be multiplied with each term in the second.

This process is often supported by the distributive property, as it helps to ensure all terms are considered. Binomial multiplication is foundational in algebra because understanding this concept helps solve more complex polynomial equations and is widely used throughout higher-level math courses.