Problem 54 - Algebra A Combined Function

Question

Multiply. $$ (x+7)(x+7) $$

Step-by-Step Solution

Answer

The solution is \(x^2 + 14x + 49\).

Step 1: Write the Expression

We start with the expression \((x+7)(x+7)\), which represents the multiplication of the binomial \((x+7)\) by itself.

Step 2: Apply the Formula for Squaring a Binomial

To multiply the expression, use the formula for squaring a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x\) and \(b = 7\).

Step 3: Calculate \(a^2\)

Find \(a^2\) where \(a = x\), so \( a^2 = x^2 \).

Step 4: Calculate \(2ab\)

Calculate \(2ab\) where \(a = x\) and \(b = 7\), so \(2ab = 2 \cdot x \cdot 7 = 14x\).

Step 5: Calculate \(b^2\)

Find \(b^2\) where \(b = 7\), so \(b^2 = 7^2 = 49\).

Step 6: Combine the Results

Using the results from Steps 3, 4, and 5, the expression becomes \(x^2 + 14x + 49\).

Key Concepts

Squaring a Binomial

Understanding how to square a binomial is a fundamental skill in algebra. Consider the expression \(x + 7\), which is our binomial. When we square this binomial, we multiply it by itself: \( (x+7)(x+7) \).

Squaring a binomial follows a simple formula: \( (a + b)^2 = a^2 + 2ab + b^2 \). This formula helps us break down the expression into manageable parts. For our example, \(a\) is \(x\), and \(b\) is \(+7\).

When we apply this formula, we find three components:

• First, compute \(a^2\), which is \(x^2\).
• Next, find \(2ab\), calculated as \(2 \cdot x \cdot 7 = 14x\).
• Finally, calculate \(b^2\), which is \(7^2 = 49\).

When combined, these terms reveal the squared binomial as \(x^2 + 14x + 49\). This formula not only simplifies computations but also lays the groundwork for understanding polynomial expansions in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent specific values or sets of values. These expressions form the basis of algebra and are used to solve equations, model real-world phenomena, and simplify complex problems.

In the expression \( (x+7)(x+7) \), \(x\) is a variable representing an unknown value. The number \(7\), a constant, is added to \(x\). Such expressions allow us to work algebraically, providing a framework for expanding, simplifying, and solving mathematical problems.

Key components of algebraic expressions include:

• Variables: Typically represented by letters such as \(x\) or \(y\).
• Constants: These are known values, like \(7\) in our example.
• Operations: Addition, subtraction, multiplication, and division can be applied.

Understanding algebraic expressions allows you to manipulate them to reach a desired outcome. This manipulation forms the bedrock of more advanced mathematical concepts.

Polynomial Multiplication

Polynomial multiplication involves expanding an expression where terms, each possibly containing variables raised to powers and multiplied by coefficients, are multiplied together. This is another key concept in algebra that often requires using formulas and techniques to simplify expressions.

To multiply polynomials, distribute each term in one polynomial to each term in the other. In our example, \( (x+7)(x+7) \), we used the special case method of squaring a binomial, which simplifies the multiplication through a specific formula.

The goal in polynomial multiplication is to combine like terms and simplify the expression to its most reduced form. Remember to:

• Use distribution carefully to ensure every term from the first polynomial multiplies with every term from the second.
• Identify like terms such as those involving the same powers and variables, and then simplify them.
• Reapply algebraic rules for addition and multiplication to finalize the expression.

Understanding these steps makes complex polynomial multiplication manageable and builds a foundation for tackling more advanced topics in algebra.