Problem 31 - Elementary Algebra Within Reach

Question

In Exercises 19-36, expand the expression as a product of factors. $$ (x+y)^{2} $$

Step-by-Step Solution

Answer

So, the expansion of \((x + y)^2\) as a product of factors is \(x^2 + 2xy + y^2 \)

Step 1: Apply the Binomial formula

First, apply the binomial formula \((a+b)^2 = a^2 + 2ab + b^2 \) to the given expression. Substitute a with x and b with y in this formula. This leads to \(x^2 + 2xy + y^2 \)

Step 2: Simplify the expression

The expression does not need further simplification, thus it becomes \(x^2 + 2xy + y^2 \)

Key Concepts

Quadratic Expansion

Quadratic expansion is a method used in algebra to expand expressions that are raised to the second power, or squared. When you encounter a term such as \((x + y)^2\), you're dealing with a classic case of a quadratic expansion.
The easiest way to handle these expressions is by using the Binomial Theorem, particularly the formula for squaring a binomial.

• The Binomial Theorem allows you to expand \((a + b)^2\) into \(a^2 + 2ab + b^2\).
• In the given problem, \(a\) is replaced by \(x\), and \(b\) is replaced by \(y\).
• By substitution, \((x + y)^2\) becomes \(x^2 + 2xy + y^2\).

Quadratic expansions are straightforward once you memorize the formula, and they allow us to convert binomial expressions into simple polynomial expressions quickly. Using these expansions is fundamental in algebra as it simplifies working with algebraic expressions.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. They can take different forms and have various degree levels, depending on the highest power of the variable in the expression.
In the expression \(x^2 + 2xy + y^2\), you have a polynomial with:

• Three terms: \(x^2\), \(2xy\), and \(y^2\).
• Each term is a product of a coefficient and variables raised to powers.
• The degree of the polynomial is 2, given by the highest sum of powers in a single term.

Polynomials like this are fundamental units in algebra, serving as building blocks for more complex expressions. Polynomials make it easier to structure and solve equations and are used in calculus, physics, and engineering problems too.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations, such as addition, subtraction, multiplication, and division. They form the basic language of algebra and allow us to use variables instead of specific numbers to solve problems. In the case of \((x + y)^2\), this expression represents a compact form that can be expanded and simplified using algebraic operations.
Key aspects of algebraic expressions include:

• Variables are symbols that represent numbers, like \(x\) and \(y\) in our example.
• Operations signify how numbers and variables interact within the expression.
• Algebraic structure allows these expressions to model real-world situations, relationships, and helps us perform computations efficiently.

Understanding algebraic expressions involves learning how to manipulate and transform them, as in the quadratic expansion task provided. Mastery of expressions is crucial for progressing in algebra and mathematics in general.