Problem 84 - Algebra A Combined Function

Question

Multiply. See Section 5.6. \((x+3)^{2}\)

Step-by-Step Solution

Answer

The expanded form is \(x^2 + 6x + 9\).

Step 1: Understand the Problem

The given problem requires expanding the expression \((x+3)^2\). This means multiplying the binomial \((x+3)\) by itself.

Step 2: Apply the Formula

Recall the formula for squaring a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). Here, \((x+3)^2\) corresponds to \((a+b)^2\) with \(a = x\) and \(b = 3\).

Step 3: Square the First Term

Calculate \(x^2\), which is the square of the first term of the binomial: \(x^2 = x \times x = x^2\).

Step 4: Calculate Twice the Product of the Terms

Calculate \(2ab\), which is twice the product of the terms: \(2 \times x \times 3 = 6x\).

Step 5: Square the Second Term

Calculate \(b^2\), which is the square of the second term of the binomial: \(3^2 = 9\).

Step 6: Combine All Parts Together

Add the results from the previous steps: \(x^2 + 6x + 9\). This is the expanded form of \((x+3)^2\).

Key Concepts

Squaring a Binomial

Squaring a binomial is a fundamental algebraic process that expands an expression of the form \((a+b)^2\). Essentially, you are multiplying the binomial by itself, which follows a straightforward pattern. The formula \((a+b)^2 = a^2 + 2ab + b^2\) simplifies this process. Here, \(a\) and \(b\) represent any two terms in the binomial.

To square a binomial like \((x+3)^2\), identify the parts corresponding to \(a\) and \(b\). In this example, \(a = x\) and \(b = 3\).

• First: Square the first term, \(a^2\). Hence, \(x^2\).
• Second: Calculate twice the product of the first and second terms, or \(2ab\). Here, \(2 \times x \times 3 = 6x\).
• Third: Square the second term, \(b^2\), resulting in \(3^2 = 9\).

Finally, combine these parts: \(x^2 + 6x + 9\). This formula helps you see the expanded form of a squared binomial, breaking down the steps into manageable chunks.

Distributive Property

The distributive property is a powerful tool in algebra that allows you to simplify expressions by distributing multiplication over addition. For squaring a binomial, the distributive property helps explain why the binomial formula works.

When you expand \((x+3)(x+3)\), use the distributive property:

• First, distribute \(x\) from the first binomial to both terms in the second, obtaining \(x \times x + x \times 3\).
• Then, distribute \(3\) from the first binomial to both terms in the second, resulting in \(3 \times x + 3 \times 3\).

Bringing it all together, the combinations yield: \(x^2 + 3x + 3x + 9\).

Combine like terms, such as \(3x + 3x\) to get \(6x\). The final expanded form, \(x^2 + 6x + 9\), matches the result from the binomial formula. The distributive property ensures each piece of the expression is multiplied correctly, leading to an accurate expansion.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to achieve a new expression. A common scenario occurs when squaring binomials, like \((x+3)^2\), essentially a special case of polynomial multiplication. Let's break down how this multiplicative process works step-by-step.

To multiply \((x+3)\) by itself, consider the following:

• Recognize that \((x+3)\) is a polynomial with two terms.
• Apply the multiplication logically, multiplying each term of the first polynomial by each term of the second.

In our case, multiply each component individually:

• Multiply the first terms: \(x \times x = x^2\).
• Multiply the outer terms: \(x \times 3 = 3x\).
• Multiply the inner terms: \(3 \times x = 3x\).
• Multiply the last terms: \(3 \times 3 = 9\).

Collectively, these individual products compose the expression: \(x^2 + 3x + 3x + 9\). Simplifying like terms results again in \(x^2 + 6x + 9\). Understanding each multiplication step clarifies how expanding polynomials works in algebraic contexts.