Problem 38 - Elementary and Intermediate Algebra

Question

Expand each binomial. See Example 3. $$ (y+2)^{3} $$

Step-by-Step Solution

Answer

The expanded form is \(y^3 + 6y^2 + 12y + 8\).

Step 1: Identify the Binomial and its Exponent

The binomial is \((y + 2)\) and it is raised to the power of 3, meaning it should be expanded according to the binomial theorem.

Step 2: Apply the Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). Here, \(a = y\), \(b = 2\), and \(n = 3\).

Step 3: Calculate Each Term of the Expansion

Calculate each term using the binomial coefficients:- For \(k=0\): \( \binom{3}{0} y^{3-0} 2^0 = 1 \cdot y^3 \cdot 1 = y^3\)- For \(k=1\): \( \binom{3}{1} y^{3-1} 2^1 = 3 \cdot y^2 \cdot 2 = 6y^2\)- For \(k=2\): \( \binom{3}{2} y^{3-2} 2^2 = 3 \cdot y^1 \cdot 4 = 12y\)- For \(k=3\): \( \binom{3}{3} y^{3-3} 2^3 = 1 \cdot 1 \cdot 8 = 8\)

Step 4: Combine All Terms

Combine all the calculated terms to form the expanded expression: \( y^3 + 6y^2 + 12y + 8 \).

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in algebra that provides a formula to expand expressions raised to any power. It's particularly useful for expressions of the form

• \((a + b)^n\)

where \(a\) and \(b\) are any numbers or variables, and \(n\) is a positive integer.

The theorem states that such an expression can be expanded into a sum of terms involving binomial coefficients. Mathematically, this looks like:

• \[(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]

where \(\binom{n}{k}\) are the binomial coefficients, which count the number of ways to choose \(k\) elements from a set of \(n\) elements. For example, when expanding \((y + 2)^3\), the binomial coefficients are 1, 3, 3, and 1, corresponding to the terms in the expansion: \(y^3\), \(6y^2\), \(12y\), and 8.

The binomial theorem not only saves time but also helps identify symmetric properties of polynomials. Mastery of this theorem is a key step in understanding more complex algebraic expressions.

Polynomial Expansion

Polynomial expansion is the process of rewriting an expression represented by a polynomial in a more extended form. This is what happens when a binomial, such as

• \((y + 2)^3\)

is expanded using algebraic methods like the binomial theorem.

Expanding a polynomial involves multiplying the terms and using the properties of exponents and coefficients.

• For each term \((k)\) in the expansion, calculate all possible products of \(y\) and \(2\) raised to their respective powers.

This creates terms like \(y^3\), \(6y^2\), \(12y\), and 8. Polynomial expansion allows complex expressions to be broken down into simpler terms, which can then be easily added up or rearranged based on the requirements.

Understanding polynomial expansion is crucial in algebra as it helps solve equations, integrate functions, and even in calculus applications where polynomial approximations are needed.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and algebraic operations like addition, subtraction, multiplication, and division. When dealing with binomials such as

• \((y + 2)\)

it's important to understand how these elements interact to form the expanded expression of a polynomial.

In the expression \((y + 2)^3\),

• \(y\) and \(2\) are added together and then raised to the power of 3.

The expansion then involves distributing the power across each term within the parentheses and utilizing algebraic rules to combine like terms.

Algebraic expressions like these enable us to model and solve real-world problems. By manipulating and transforming the expressions, one can discern patterns, solve equations, and simplify complex problems. This understanding forms the foundation for higher-level mathematics and is applied extensively in various scientific and engineering fields.