Problem 49 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$\left(\frac{1}{3} y+2 z^{2}\right)^{3}$$

Step-by-Step Solution

Answer

The expanded form of the expression \(\left(\frac{1}{3} y+2 z^{2}\right)^{3}\) using the binomial theorem is: \(\frac{1}{27}y^3 + \frac{2}{3}y^2z^2 + 4yz^4 + 8z^6\).

Step 1: Identify the terms and degree of the expansion

In the expression \(\left(\frac{1}{3} y+2 z^{2}\right)^{3}\), we have \(\frac{1}{3}y\) and \(2z^2\) as our terms (a and b), and the degree (n) of the expansion is 3. So, we will apply the binomial theorem to obtain the expanded form of the given expression.

Step 2: Apply the binomial theorem formula

Using the binomial theorem formula, we have: \((a + b)^3 = \displaystyle\sum_{k=0}^3 {3 \choose k}a^{3-k}b^k\) Substitute the values of a and b into the formula: \(\left(\frac{1}{3}y + 2z^2\right)^3 = \displaystyle\sum_{k=0}^3 {3 \choose k}\left(\frac{1}{3}y\right)^{3-k}(2z^2)^k\)

Step 3: Expand and simplify the expression

Now we can expand and simplify the expression: \(= {3 \choose 0}\left(\frac{1}{3}y\right)^3(2z^2)^0 + {3 \choose 1}\left(\frac{1}{3}y\right)^2(2z^2)^1 + {3 \choose 2}\left(\frac{1}{3}y\right)^1(2z^2)^2 + {3 \choose 3}\left(\frac{1}{3}y\right)^0(2z^2)^3\) Now, find the binomial coefficients: \(= \frac{3!}{0!(3-0)!}\left(\frac{1}{3}y\right)^3(2z^2)^0 + \frac{3!}{1!(3-1)!}\left(\frac{1}{3}y\right)^2(2z^2)^1 + \frac{3!}{2!(3-2)!}\left(\frac{1}{3}y\right)^1(2z^2)^2 + \frac{3!}{3!(3-3)!}\left(\frac{1}{3}y\right)^0(2z^2)^3\) Next, find the factorials and simplify: \(= \frac{6}{6}\left(\frac{1}{27}y^3\right)(1) + \frac{6}{2}\left(\frac{1}{9}y^2\right)(2z^2) + \frac{6}{2}\left(\frac{1}{3}y\right)(4z^4) + \frac{6}{6}(1)(8z^6)\) \(= \left(\frac{1}{27}y^3\right) + 3\left(\frac{2}{9}y^2z^2\right) + 3\left(\frac{4}{3}y z^4\right) + (8z^6)\) Finally, simplify the coefficients: \(= \frac{1}{27}y^3 + \frac{2}{3}y^2z^2 + 4yz^4 + 8z^6\) So, the expansion of \(\left(\frac{1}{3} y+2 z^{2}\right)^{3}\) using the binomial theorem is: \(\frac{1}{27}y^3 + \frac{2}{3}y^2z^2 + 4yz^4 + 8z^6\)

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, structured as the sum of terms. Each term is a product of a number (called the coefficient) and a variable raised to a non-negative integer power. Polynomials can have one or multiple terms, depending on the situation. For example, the polynomial \(3x^2 + 2x + 1\) has three terms. In our exercise, the expression before expansion is considered a polynomial with terms such as \((\frac{1}{3} y)^3\) and \(2z^{2}\).

Key features of polynomials include:

• Degree: This is the highest power of the variable in the polynomial. It determines the polynomial's behavior for large values.
• Standard Form: A polynomial is usually expressed in descending powers of the variable.
• Leading Coefficient: The coefficient of the first term in standard form.

Understanding polynomials is crucial because they form the basis for a wide range of topics in algebra and other areas of mathematics.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). Unlike equations, these expressions don’t have an "equals" sign. Instead, they describe a value or range of values.

In the expression \((\frac{1}{3}y + 2z^2)^3\), \(\frac{1}{3}y\) and \(2z^2\) are parts of an algebraic expression. Here, "\(+\)" is the operation connecting the two parts. The expression becomes more complex when raised to a power because it involves multiple components interacting with each other through multiplication and distribution.

• Simplification: The goal with algebraic expressions often involves simplifying them by combining like terms or using special rules like the distributive property.
• Factorization: Sometimes, we rewrite expressions as products of simpler expressions, useful for solving equations.

Understanding algebraic expressions is essential for not only solving equations but also for exploring various mathematical models and scenarios.

Mathematical Expansion

Mathematical expansion is a process of expressing a power expression, such as \((a+b)^n\), in a detailed form composed of individual terms. This process often relies on special mathematical tools like the Binomial Theorem.

The Binomial Theorem provides us with a way to expand expressions of the form \((a + b)^3\) into a sum of terms, which involves using values called binomial coefficients. These coefficients are determined by the combinations formula \(\binom{n}{k}\), where \(n\) is the degree and \(k\) is an index that varies from 0 up to \(n\). The expansion of \((\frac{1}{3}y + 2z^2)^3\) demonstrates how each term of the original expression contributes to the final expanded polynomial.

• Pattern Recognition: By expanding expressions, we can observe and utilize patterns that make calculation easier.
• Applications: Expansion is useful in calculus and algebra for evaluating complex expressions and simplifying calculations.

Mathematical expansion is not just a theoretical exercise but a practical tool for handling complex problems across numerous mathematical fields.