Problem 70 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (x+7 y)^{2} $$

Step-by-Step Solution

Answer

The simplified product of \((x+7y)^2\) is \(x^2 + 14xy + 49y^2\).

Step 1: Identify the terms

We have a binomial (x+7y), and we are square this binomial. The first term \(a\) is \(x\) and the second term \(b\) is \(7y\).

Step 2: Apply the binomial square formula

Plug the terms \(a\) and \(b\) into the binomial square formula \((a+b)^2 = a^2 + 2ab + b^2\). We get \((x+7y)^2 = x^2 + 2*x*7y + (7y)^2\).

Step 3: Simplify the equation

Simplify the step 2 equation. The result will be \(x^2 + 14xy + 49y^2\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a foundational concept in mathematics. They consist of numbers, variables, and operations combined together. In the expression \((x + 7y)^2\), \(x\) and \(y\) are variables, while 7 is a coefficient.

Expressions like this can include:

• Variables: Symbols representing numbers (e.g., \(x, y\)).
• Constants: Fixed numbers (e.g., 7, 49).
• Operators: Symbols indicating operations (e.g., +, -, *).

An algebraic expression doesn't have an equal sign. It represents a value that can change depending on the variables. Understanding how to manipulate these expressions is key to solving algebraic problems.

Polynomials

Polynomials are a specific type of algebraic expression that include terms of variables raised to whole number powers. In our example, \(x^2 + 14xy + 49y^2\), each term is a part of the polynomial.

A polynomial consists of:

• Terms: Parts of the expression separated by + or -.
• Coefficients: Numbers multiplying the variables (e.g., 14 in 14xy).
• Degree: The highest power of the variable (e.g., 2 in \(x^2\)).

Polynomials can be classified by degree and the number of terms they have, such as monomials (1 term), binomials (2 terms), and trinomials (3 terms). Understanding polynomials is crucial for exploring more complex algebraic operations.

Distributive Property

The distributive property is a helpful tool for expanding expressions. It allows us to multiply a single term by each term in a parenthesis. This property is used in binomial expansion.

With our expression \((x+7y)^2\), we apply the distributive property using the formula \((a+b)^2 = a^2 + 2ab + b^2\).

• Multiply each term separately: \(x^2\), \(2 \times x \times 7y\), \((7y)^2\).
• Combine the results: \(x^2 + 14xy + 49y^2\).

The distributive property simplifies the process of multiplying and transforming complex expressions into simpler ones, making solving algebraic problems easier.