Problem 42 - Precalculus with Limits

Question

In Exercises 41 - 44, expand the binomial by using Pascals Triangle to determine the coefficients \( \left(3 - 2z\right)^4 \)

Step-by-Step Solution

Answer

The expanded form of \((3 - 2z)^4\) using Pascal's Triangle for the coefficients is \(81 - 108z + 216z^2 - 96z^3 + 16z^4\).

Step 1: Understanding Pascal's Triangle

Pascal's Triangle is a geometric representation of the coefficients of binomial expansion. The 4th row (we start counting from 0) of Pascal's Triangle is '1 4 6 4 1', which represents the coefficients of the binomial \((a + b)^4\).

Step 2: Applying the Coefficients

Applying the coefficients from Pascal's Triangle to the expansion \((3 - 2z)^4\), we get: \[1*(3^4)*(-2z)^0 - 4*(3^3)*(-2z)^1 + 6*(3^2)*(-2z)^2 - 4*(3^1)*(-2z)^3 + 1*(3^0)*(-2z)^4\]. Notice how the decreasing powers of the first term and increasing powers of the second term in the binomial always sum to 4, which is the original power of the binomial. Also note the alternating signs, which comes from the minus sign in the binomial.

Step 3: Simplifying and finalizing the expansion

On simplifying the expression, we get: \[81 - 4*27z + 6*36z^2 - 4*24z^3 + 16z^4 = 81 - 108z + 216z^2 - 96z^3 + 16z^4\]. So, the expanded form of \((3 - 2z)^4\) is \(81 - 108z + 216z^2 - 96z^3 + 16z^4\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a fascinating and helpful tool in mathematics that simplifies the process of finding coefficients in binomial expansion. Each row of Pascal's Triangle corresponds to the coefficients of the expanded form of a binomial raised to a power. Row numbers start from zero. So, for example, row 4, which is '1 4 6 4 1', defines the coefficients for \((a + b)^4\). It helps us predict how terms in a binomial will expand without lengthy calculations. Simply pick a row corresponding to the binomial's power to find the coefficients you need.

Coefficients

Coefficients are the numbers that multiply expressions and play a key role in polynomial expansion. In algebra, particularly when dealing with binomials, these numbers tell us how terms distribute across the power of a binomial expression.
When we expanded \((3 - 2z)^4\), we took our coefficient values directly from Pascal's Triangle, which were 1, 4, 6, 4, 1. These coefficients multiply each term of the binomial expansion as each variable is raised to decreasing and increasing powers.

Polynomial Expansion

Polynomial expansion involves stretching out an expression like \((a + b)^n\). It means writing the expression as a series of terms, each accompanied by a coefficient, making calculations easier to manage.
For example, expanding \((3 - 2z)^4\) requires multiplying the coefficients by positive and negative powers, and simplifying where necessary. This results in a sum of individual polynomial terms, as shown in 81 - 108z + 216z^2 - 96z^3 + 16z^4.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra and simplify problem-solving by providing a concise way to express calculations.
In the binomial expansion of \((3 - 2z)^4\), each term is a separate algebraic expression. These expressions include both constants and variables, correctly informing us about how the values interact through addition and subtraction, which helps in deriving meaningful solutions from what initially seems complex.