Problem 29 - Algebra 2

Question

Expand each binomial. $$ \left(x^{2}+y^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The expansion of the binomial \((x^{2}+y^{2})^{2}\) is \(x^4 + 2x^2y^2 + y^4\)

Step 1: Consider the algebraic expression

Begin with the expression \((x^{2}+y^{2})^{2}\). In this case, \(a = x^2\) and \(b = y^2\).

Step 2: Plug into the formula

Plug \(a\) and \(b\) into the formula \((a + b)^2 = a^2 + 2ab + b^2\). This gives you \((x^{2} + y^{2})^{2} = (x^2)^2 + 2 * x^2 * y^2 + (y^2)^2\).

Step 3: Simplify the expression

Simplify the expression to obtain the final answer. \((x^2)^2\) simplifies to \(x^4\), \(2 * x^2 * y^2\) remains the same, and \((y^2)^2\) simplifies to \(y^4\).

Step 4: Formulate the final expression

Combine all the terms to write down the final expanded expression, which is \(x^4 + 2x^2y^2 + y^4\)

Key Concepts

Polynomial Expressions

Polynomials are fundamental expressions in algebra that consist of variables and coefficients, combined using addition, subtraction, and multiplication. The term 'polynomial' comes from "poly," meaning many, and "nomial," which implies terms.
A polynomial expression can take on many forms, but often follows the structure:

• A polynomial like \(x^2 + 2x + 1\) involves three terms where each part is built from a different power of the variable \(x\).
• The highest exponent of the variable in the polynomial determines its degree. In the previous example, \(x^2\) is the highest power, making it a degree 2 polynomial.
• In the exercise, the expression \((x^2 + y^2)^2\) involves polynomials within polynomials, showcasing how complex expressions can be constructed and then simplified through algebraic processes like expansion.

Expanding polynomial expressions is a crucial skill in algebra, aiding in solving equations and understanding functions in higher mathematics.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are the language of algebra, allowing us to represent real-world problems in a mathematical context.

• In our exercise, \((x^2 + y^2)^2\) is an algebraic expression where the binomial is raised to a power, demanding a method like binomial expansion for simplification.
• Algebraic expressions are simplified by combining like terms, using distribution, and applying operations like factoring or expansion to rewrite them in a more manageable form.
• Working with these expressions helps develop skills necessary for transitioning into solving equations, modeling situations, and finding function values.

When expanding or simplifying algebraic expressions, understanding each step, such as identifying parts of the binomial \(a\) and \(b\) in formulas, is essential to achieve correct results.

Exponents

Exponents are a way to express repeated multiplication of a number by itself, streamlining complex arithmetical operations. In algebra, they are denoted by a small number placed to the upper right of the base value.

• The expression \(x^n\) shows that the base \(x\) is multiplied by itself \(n\) times.
• In the example \((x^2 + y^2)^2\), exponents dictate that each term within the parenthesis is treated individually during expansion, following mathematical laws of distributing powers over sums.
• For expanding, knowing that \((a + b)^2 = a^2 + 2ab + b^2\) utilizes the properties of exponents and binomials.

Using exponents in expressions simplifies lengthy multiplications and helps with identifying how variables interact through operations. Exponents also bridge to more advanced topics like logarithms and exponential functions, foundational in higher-level mathematics.