Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, expand the binomial \((3 y-6)^{2}\)

Step-by-Step Solution

Answer

The expanded form of \((3y - 6)^2\) is \(9y^2 - 36y + 36\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \((a + b)^n\) can be expanded using the formula \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, we need to expand \((3y - 6)^2\). This is a special case where \(n = 2\).

Step 2: Identify Variables

Identify the components of the binomial: let \(a = 3y\) and \(b = -6\). The exponent \(n = 2\).

Step 3: Apply the Binomial Expansion Formula

Use the Binomial Theorem for \((a-b)^2\) which is \(a^2 - 2ab + b^2\). Here, \(a = 3y\) and \(b = 6\).

Step 4: Calculate Each Term

First, calculate \(a^2 = (3y)^2 = 9y^2\). Then calculate \(-2ab = -2 \cdot 3y \cdot 6 = -36y\). Finally, calculate \(b^2 = 6^2 = 36\).

Step 5: Combine the Terms

Combine these results to construct the expanded form. Thus, \((3y - 6)^2 = 9y^2 - 36y + 36\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions raised to a power. Imagine you have a binomial, which is an algebraic expression containing two terms, like \((a + b)^n\), where \(a\) and \(b\) can be any numbers or variables, and \(n\) is a positive integer. The theorem provides a systematic way to expand such expressions without manual multiplication. This is done using the formula: \[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
This formula utilizes binomial coefficients \(\binom{n}{k}\), which you might recognize as entries in Pascal's Triangle, to find the coefficients of the expanded terms. When the binomial is raised to the 2nd power, it's particularly simple. For example, applying this theorem to \((3y - 6)^2\), allows us to systematically find each term of the expansion, reducing the chances of errors that might happen with simple distributive multiplication.

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of variables with coefficients, like \(ax^n + bx^{n-1} + cx^{n-2} + ...\), where the powers are non-negative integers. They form the building blocks of a lot of algebraic manipulation and appear commonly in calculus and other areas of mathematics. A polynomial of degree two, like the expanded form of our binomial \((3y - 6)^2 = 9y^2 - 36y + 36\), is particularly known as a quadratic polynomial.
Quadratics play a crucial role in various applications, from physics to economics, where they describe parabolic relations. By understanding and expanding binomials, we gain the ability to work with polynomials more effectively. Once expanded, the polynomial can be more easily analyzed for its characteristics such as roots, vertex, and intercepts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication) that represent a particular value. In our exercise, simplifying \((3y - 6)^2\) into \(9y^2 - 36y + 36\) transforms the expression into a simpler, more understandable form. Algebraic expressions are foundational in mathematics because they provide a way to describe mathematical relationships and operations clearly and concisely.
When working with algebraic expressions, becoming familiar with manipulating variables and constants is crucial. This includes understanding how to add, subtract, multiply, and divide these expressions. Knowing how to properly handle expressions leads to a better grasp of more complicated algebraic concepts, ensuring you can solve equations, simplify expressions, and calculate effectively.