Problem 9 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (x+2)(x+4) $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: \((x+2)(x+4)\) Answer: The simplified algebraic expression is \(x^2 + 6x + 8\).

Step 1: Apply the distributive property

To simplify \((x+2)(x+4)\), we need to multiply each term in the first parenthesis with each term in the second parenthesis. This means that we will have four multiplications in total, which are: 1. \(x \times x\) 2. \(x \times 4\) 3. \(2 \times x\) 4. \(2 \times 4\)

Step 2: Perform the multiplications

Now, we will perform the multiplications from Step 1 and obtain the following results: 1. \(x \times x = x^2\) 2. \(x \times 4 = 4x\) 3. \(2 \times x = 2x\) 4. \(2 \times 4 = 8\)

Step 3: Combine like terms

After performing the multiplications in Step 2, we have the terms \(x^2\), \(4x\), \(2x\), and \(8\). To simplify the expression, we can combine the like terms (the terms that involve the same power of \(x\)). In this case, we can combine the terms \(4x\) and \(2x\) to get a single term involving \(x\): \(4x + 2x = 6x\).

Step 4: Write the simplified expression

Finally, we will write the simplified algebraic expression by combining all the terms: \(x^2 + 4x + 2x + 8\) is equivalent to \(x^2 + 6x + 8\). Thus, the simplified algebraic expression is \((x+2)(x+4) = x^2 + 6x + 8\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to multiply a term across a set of terms inside parentheses. When you're faced with an expression like

• \((x+2)(x+4)\), apply the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis.
• This means multiplying:
  • \(x\) by \(x\), which gives \(x^2\)
  • \(x\) by \(4\), resulting in \(4x\)
  • \(2\) by \(x\), producing \(2x\)
  • \(2\) by \(4\), giving \(8\)

This process is key to breaking down and simplifying expressions by replacing complex binomials into simpler, manageable terms through multiplication.
Learning to work with the distributive property helps in practically every algebraic manipulation involving expressions and polynomials.

Like Terms

Like terms are terms in algebraic expressions that have the same variable raised to the same power. You can combine these terms to simplify expressions. Identifying like terms is crucial.
For the expression we are working with:

• After applying the distributive property and deriving the terms \(x^2\), \(4x\), \(2x\), and \(8\), notice that \(4x\) and \(2x\) are like terms.
• Both contain the variable \(x\) raised to the first power.

To simplify the expression, add the coefficients of like terms together:

• \(4x + 2x\) becomes \(6x\).

Recognizing and combining like terms reduces complexity, making expressions easier to understand and work with.

Simplification of Polynomials

Simplifying polynomials involves rewriting them in a more compact and readable form. It typically requires applying the distributive property, combining like terms, and arranging terms in descending order of their exponents.
In our example \((x+2)(x+4)\):

• First, use the distributive property to expand the expression.
• Then, combine the like terms \(4x + 2x\) to simplify \(6x\).
• Finally, assemble the simplified form: \(x^2 + 6x + 8\).

By following these steps, you've transformed a product of binomials into a simplified polynomial. Simplified expressions are easier to interpret and provide a clearer understanding of the relationships between variables.