Problem 12 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$(x+3 y)^{3}$$

Step-by-Step Solution

Answer

The polynomial is \(x^3 + 9x^2 y + 27xy^2 + 27y^3\).

Step 1: Apply the Binomial Theorem

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = x\), \(b = 3y\), and \(n = 3\). We will expand \((x + 3y)^3\) using this formula.

Step 2: Calculate Each Term Independently

Substitute into the Binomial Theorem: \[(x + 3y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} (3y)^k\]This gives us individual terms: - For \(k = 0\): \(\binom{3}{0} x^3 (3y)^0 = x^3\) - For \(k = 1\): \(\binom{3}{1} x^2 (3y)^1 = 3x^2 \cdot 3y = 9x^2 y\) - For \(k = 2\): \(\binom{3}{2} x^1 (3y)^2 = 3x \cdot 9y^2 = 27x y^2\) - For \(k = 3\): \(\binom{3}{3} x^0 (3y)^3 = 27y^3\)

Step 3: Sum All Calculated Terms

Now, add all the terms from Step 2 to express the polynomial:\[x^3 + 9x^2 y + 27xy^2 + 27y^3\].This polynomial is the final expanded form of \((x + 3y)^3\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra. It involves transforming an expression with exponents into a series of terms. In our example, we expand \((x+3y)^{3}\) into a polynomial. This means we will rewrite it as a sum of multiple terms without exponents. Polynomial expansion is especially useful for simplifying expressions and solving equations.

• The process uses rules and formulas, such as the Binomial Theorem, to break down complex expressions into simpler parts.
• This transformation helps us to better understand the structure and function of algebraic expressions.

Expanding a polynomial allows us to see every contribution to the final form of the expression.

Binomial Coefficients

Binomial coefficients play a key role in expanding expressions like \((x+3y)^3\). These coefficients are the numbers that multiply the terms in the polynomial. They arise from the Binomial Theorem, represented by \(\binom{n}{k}\), where \(n\) is the total exponent and \(k\) is the specific term index.

• The binomial coefficient tells us how many ways we can choose \(k\) items from \(n\) items, which directly influences the terms' multiplication factor during expansion.
• For example, the coefficients for \((x+3y)^{3}\) are found from \(\binom{3}{k}\) for \(k = 0, 1, 2, 3\).
• This results in coefficients 1, 3, 3, and 1, which are used to scale the corresponding terms \(x^{3-k} (3y)^k\).

Understanding binomial coefficients is essential as they determine the balance and symmetry in polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of algebra and include terms like \(x, 3y,\) and \(x^3\). In our exercise, \((x+3y)^3\) is expanded into another algebraic expression: \(x^3 + 9x^2 y + 27xy^2 + 27y^3\).

• Each term in the expression represents a distinct component that contributes to the expression's value.
• By manipulating variables and coefficients, algebraic expressions can be added, subtracted, multiplied, and divided.
• When expanding \((x+3y)^3\), each term, like \(9x^2y\), results from combining coefficients with powers of \(x\) and \(y\).

Grasping algebraic expressions equips you with the tools needed to construct, deconstruct, and simplify mathematical problems.