Problem 62 - Precalculus with Limits

Question

In Exercises 61 - 66, use the Binomial Theorem to expand and simplify the expression. \( \left(2\sqrt{t} - 1\right)^3 \)

Step-by-Step Solution

Answer

The expanded and simplified form of \( \left(2\sqrt{t} - 1\right)^3 \) is \( 8t\sqrt{t} - 12t + 6\sqrt{t} - 1 \).

Step 1: Understand the Binomial Theorem

The expanded form of \( (a-b)^n \), where n is a positive integer, according to the Binomial Theorem is \( a^n - na^{n-1}b + \frac{n(n-1)}{2!} a^{n-2}b^2 - ... + (-1)^nb^n \). Here, a and b are any real or complex numbers, and n must be a positive integer.

Step 2: Apply the Binomial Theorem

The given expression is \( \left(2\sqrt{t} - 1\right)^3 \). Let's liken it to the binomial theorem expression: Here, \( a = 2\sqrt{t} \), \( b = 1 \), and \( n = 3 \). Following the formula, \( (a-b)^n = a^n - na^{n-1}b + \frac{n(n-1)}{2!} a^{n-2}b^2 - (-1)^nb^n \). Substitute a, b, and n in the formula, which results in: \( (2\sqrt{t} - 1)^3 = (2\sqrt{t})^3 - 3(2\sqrt{t})^2 + 3(2\sqrt{t}) -1^3 \).

Step 3: Simplify the Expression

The expanded form from step 2 simplifies to: \( 8t\sqrt{t} - 12t + 6\sqrt{t} - 1 \).

Key Concepts

Polynomial Simplification

Polynomial simplification is a process used in algebra to reduce complex polynomials to their simplest form. This process often involves combining like terms and following the order of operations. For instance, in the solution to the problem, the expression \( (2\sqrt{t} - 1)^3 \) is expanded using the Binomial Theorem, which generates terms that are initially presented as powers of a binomial.

The simplification step involves rewriting each term to its simplest form; for example, simplifying \( (2\sqrt{t})^3 \) to \( 8t\sqrt{t} \) by exponentiating each component of the first term. Additionally, combining like terms and constants helps in obtaining the more manageable and representable equation. In complex cases, factoring, dividing polynomials, or using synthetic division might be needed for simplification. To ensure the process is done correctly, remember to perform operations within brackets first, combine like terms, and systematically work through the expression.

It's critical for students to understand each of these steps because it allows for accuracy in polynomial simplification and sets a foundation for solving more advanced algebraic problems.

Binomial Coefficients

Binomial coefficients are integral to understanding the Binomial Theorem's expansion. These coefficients are the numbers that appear as multipliers of the terms in the expansion of a binomial power, represented by \( \binom{n}{k} \) (read as 'n choose k'), where \( n \) is the power, and \( k \) is the specific term's index.

In the provided exercise, to expand \( (2\sqrt{t} - 1)^3 \) using the Binomial Theorem, we calculate the coefficients for each term. This is done by following Pascal's triangle or using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( ! \) denotes the factorial of a number. For the cubic term \( (a-b)^3 \) we have the sequence of binomial coefficients 1, 3, 3, 1. These numbers efficiently dictate the number of ways to choose \( k \) elements from a set of \( n \) without regard to the order.

Grasping the concept of binomial coefficients is fundamental for students to successfully apply the Binomial Theorem and will also be beneficial in understanding combinations and probability.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, operators, and variables, but do not have an equality sign as equations do. In the solution we're discussing, \( (2\sqrt{t} - 1)^3 \) is a classic example of an algebraic expression. The process of expanding it requires knowing the order and structure of operations - from exponentiation to multiplication, and finally, addition or subtraction.

The significance of understanding algebraic expressions cannot be overstated. They are the foundation of algebra and are instrumental in defining functions, formulating equations, and solving problems across various fields of mathematics. Learning to manipulate these expressions allows students to develop crucial problem-solving skills. This includes knowing how to handle and simplify powers, as seen with \( (2\sqrt{t})^3 \) or how to distribute a negative sign throughout the terms after expanding a binomial expression. Mastery in algebraic expressions aids in transitioning to more complex concepts such as equation solving and calculus.

In conclusion, the importance of recognizing and correctly manipulating algebraic expressions lies in its frequent application within mathematics and the broader STEM fields.