Problem 58 - College Algebra

Question

Find each product. $$ (2 x-3)^{3} $$

Step-by-Step Solution

Answer

The expansion of \((2x - 3)^{3}\) is \(8x^{3} - 54x^{2} + 162x - 27\).

Step 1: Understand The Binomial Theorem

The binomial theorem is given as \((a + b)^{n} = a^{n} + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^{2} + ... + b^{n}\). Here n represents the power, 'a' and 'b' are elements of the binomial. In our problem, a = 2x, b = -3 and n = 3.

Step 2: Apply The Binomial Theorem

We substitute a = 2x, b=-3 and n=3 in the binomial theorem. So we get \((2x - 3)^{3} = (2x)^{3} + \binom{3}{1}(2x)^{2}*(-3) + \binom{3}{2}(2x)*(-3)^{2} + (-3)^{3}\)

Step 3: Calculate the binomial coefficients

\binom{3}{1} = 3, \binom{3}{2} = 3. Substitute these values into the equation: \((2x - 3)^{3} = (2x)^{3} + 3*(2x)^{2}*(-3) + 3*(2x)*(-3)^{2} + (-3)^{3}\)

Step 4: Simplify the expression

Simplify each term to get: \((2x - 3)^{3} = 8x^{3} - 54x^{2} + 162x - 27\)

Key Concepts

Polynomial Expansion

Polynomial expansion is a straightforward yet powerful technique used in algebra to express a polynomial raised to a power as a sum of terms. Each term in the expansion incorporates coefficients and the elements raised to specific powers. In our original exercise, we aimed to expand the cubic expression \[(2x - 3)^3\]This means we want to expand it into a form that consists of a sum of terms. By doing this, each term will involve powers of both '2x' and '-3'. The polynomial expansion helps to simplify complex polynomial expressions, making them easier to work with in various algebraic operations. To achieve this, we rely on the Binomial Theorem, which acts like a formula for expansion.

Binomial Coefficients

The binomial coefficients are crucial components in the binomial theorem. They are represented in mathematics as \(\binom{n}{k}\), where 'n' is the power and 'k' is the number of the term in the expansion.These coefficients essentially tell us how many ways we can choose 'k' elements from a set of 'n' elements, making them vital for proper expansion. In our exercise:

• We had \(\binom{3}{1} = 3\), indicating three ways to choose 1 item from 3.
• Similarly, \(\binom{3}{2} = 3\) also simplifies to 3, illustrating balance in these selections.

These binomial coefficients are multiplied with corresponding terms of the expansion. They are calculated using combination formulas and can also be found in Pascal's Triangle, a handy tool for quick reference. Understanding these coefficients ensures our polynomial expansion is accurate and aligns with mathematical principles.

Cubic Expressions

Cubic expressions are polynomials where the highest degree of any variable possesses a power of 3. This is evident in our exercise where we have:\[(2x - 3)^3\]It involves cubing the binomial expression. To handle cubics effectively, the Binomial Theorem simplifies taking cumbersome multiplication into a series of steps with specific coefficients leading to a simplified form.In the step-by-step solution, after correctly applying the formula, we simplified each part of the expression:

• Start by cubing \((2x)^3\) resulting in \(8x^3\).
• The next term was obtained by \(3\times (2x)^2\times (-3)\), simplified to \(-54x^2\).
• The following term simplified from \(3\times (2x)\times(-3)^2\) to \(162x\).
• The last term is \((-3)^3 = -27\).

Thus, following the simplification, the expansion yielded is:\[8x^3 - 54x^2 + 162x - 27\].Clearly, understanding the nature of cubic expressions and their expansion is essential in solving polynomial equations efficiently.