Problem 17 - College Algebra with Modeling and Visualization

Question

Use the binomial theorem to expand each expression. $$ (x+y)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(x^2 + 2xy + y^2\).

Step 1: Understand the Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(\binom{n}{k}\) denotes a binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\).

Step 2: Identify Parameters

For the expression \((x+y)^2\), the parameters are: - \(a = x\)- \(b = y\)- \(n = 2\).

Step 3: Calculate Binomial Coefficients

Calculate the binomial coefficients for \(n = 2\):- For \(k = 0\), \(\binom{2}{0} = 1\).- For \(k = 1\), \(\binom{2}{1} = 2\).- For \(k = 2\), \(\binom{2}{2} = 1\).

Step 4: Apply the Binomial Theorem

Substitute the parameters and coefficients into the binomial theorem:\[(x+y)^2 = \binom{2}{0}x^{2-0}y^0 + \binom{2}{1}x^{2-1}y^1 + \binom{2}{2}x^{2-2}y^2\]This simplifies to:\[x^2 + 2xy + y^2\]

Step 5: Write the Final Expanded Form

The expanded form of \((x+y)^2\) using the binomial theorem is:\(x^2 + 2xy + y^2\).

Key Concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving simple equations to studying abstractions such as groups, rings, and fields. In algebra, we often work with expressions that involve variables (like \(x\) and \(y\)) and constants, performing operations like addition, subtraction, multiplication, and division. These operations allow us to rearrange equations and expressions to solve for unknowns or simplify more complex problems.

• Variables: Symbols that represent numbers or values.
• Constants: Known values that do not change.
• Expressions: Combinations of variables and constants connected by operators.

Algebra is powerful because it lets us generalize mathematical problems and reasoning to find solutions for broader classes of problems. This flexibility makes it a crucial tool for both theoretical mathematics and practical problem-solving in the real world.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. They are constructed using the operations of addition, subtraction, multiplication, and non-negative integer exponents on variables. The binomial expression \((x+y)^2\) is a simple example of a polynomial.
Polynomials can be categorized based on their degree: the highest power of the variable in the expression. For instance, \((x+y)^2\) is a polynomial of degree 2 since the highest power of both variables is 2.
The terms "monomial," "binomial," and "trinomial" refer to polynomials with one, two, and three terms, respectively.

• Monomial: A single term like \(3x\).
• Binomial: Two terms like \(x + 5\).
• Trinomial: Three terms like \(x^2 + 2x + 1\).

Understanding polynomials is essential because they are foundational for many areas of mathematics and applicable to real-world phenomena, from physics to economics.

Binomial Coefficients

Binomial coefficients appear in the binomial theorem as the numbers that define how the terms in a binomial expansion are weighted. They are notated as \(\binom{n}{k}\) and represent the coefficients of the expanded terms of \((a+b)^n\). This notation shows how many ways you can choose \(k\) elements from a set of \(n\) elements without regard to order, which can be calculated using factorials:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]In the context of our expression \((x+y)^2\), the binomial coefficients determine how each term contributes to the expansion. The coefficients calculated are:

• \(\binom{2}{0} = 1\)
• \(\binom{2}{1} = 2\)
• \(\binom{2}{2} = 1\)

Each term in the expansion \(x^2 + 2xy + y^2\) corresponds to one of these coefficients, demonstrating their role in shaping the final polynomial expression. Understanding binomial coefficients also connects closely with combinatorial concepts in mathematics, helping to understand the arrangement and structure of sets.