Problem 56 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (x-1)^{3} $$

Step-by-Step Solution

Answer

The product of \( (x-1)^3 \) is \( x^3 - 3x^2 + 3x - 1 \).

Step 1: Identify the binomial components

The binomial given is \( (x-1)^3 \). The components \( a \) and \( b \) from the binomial expansion formula are, \( a=x \) and \( b=1 \).

Step2: Plug the components into the binomial cube formula

Replace \( a \) and \( b \) in the binomial cube formula \( a^3 − 3a^2b + 3ab^2 − b^3 \) with \( x \) and \( 1 \) respectively. This gives \( x^3 - 3x^2*1 + 3x*1^2 - 1^3 \).

Step 3: Simplify the expression

Calculate the expression to simplify the equation. This results in \( x^3 - 3x^2 + 3x - 1 \).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful tool in algebra that allows us to expand expressions raised to a power. It is particularly useful when the power is quite large, as it saves time and effort.

According to the binomial theorem, any binomial raised to a positive integer power can be expanded into a sum involving terms of the form \(a^n b^m\), where \(n\) is a non-negative integer and \(a\) and \(b\) are components of the binomial.

For example, for a binomial \((a+b)^n\), you can express it as a series:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Each term in the expansion involves the binomial coefficients, which are derived from Pascal's Triangle or calculated using combinations notation \(C(n, k)\).

The expansion helps in simplifying polynomial multiplication, which we will discuss next.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two polynomials together, resulting in a new polynomial. This is fundamental in algebra as it often appears in equations and real-world applications.

For the expression \((x-1)^3\), polynomial multiplication is achieved by repeating the binary operation of multiplying and adding terms. When using the binomial cube formula, the process is simplified by using previously calculated results directly.

When performing polynomial multiplication, follow these steps:

• Multiply each term in the first polynomial by each term in the second polynomial.
• Add the resulting products together.
• Combine like terms, which are terms with the same degree.

This is made easier when the expression is written in the format described through binomial expansion.

Remember, while the mechanics of multiplication may seem straightforward, managing signs, coefficients, and like terms is vital.

Binomial Cube

A specific case of using the binomial theorem is the binomial cube, where a binomial is raised to the power of three. This is a common algebraic operation with its own formula, given by:\[ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]When applying a binomial cube formula to solve \((x-1)^3\), replace \(a\) and \(b\) with \(x\) and \(-1\) respectively. The formula simplifies our calculations, providing the expanded polynomial directly.

Understanding the binomial cube:

• \(a^3\) represents multiplying \(a\) by itself three times.
• \(3a^2b\) and \(3ab^2\) represent the cross terms, where \(a\) and \(b\) are multiplied in specific orders and counted with a combinatorial factor (3 in this case).
• \(b^3\) represents \(b\) multiplied by itself three times.

The formula simplifies to \(x^3 - 3x^2 + 3x - 1\) for our example, showing the power of binomial cube expansions.