Problem 43 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(1+a^{3}\right)^{3} $$

Step-by-Step Solution

Answer

The expanded and simplified expression is \(1 + 3a^3 + 3a^6 + a^9\).

Step 1: Set Up the Problem

The problem asks us to expand and simplify the expression \((1+a^3)^3\). This means we need to expand the expression by writing it in an equivalent form where we multiply out the binomial.

Step 2: Use Binomial Theorem

To expand \((1+a^3)^3\), we apply the Binomial Theorem, which states that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k\). Here, set \(x = 1\), \(y = a^3\), and \(n = 3\). The expansion is: \[(1+a^3)^3 = \binom{3}{0}(1)^3(a^3)^0 + \binom{3}{1}(1)^2(a^3)^1 + \binom{3}{2}(1)^1(a^3)^2 + \binom{3}{3}(1)^0(a^3)^3 \]

Step 3: Calculate Binomial Coefficients

Calculate each binomial coefficient:- \(\binom{3}{0} = 1\)- \(\binom{3}{1} = 3\)- \(\binom{3}{2} = 3\)- \(\binom{3}{3} = 1\)

Step 4: Expand the Expression

Use the coefficients and powers from Step 2:- First term: \(\binom{3}{0}(1)^3(a^3)^0 = 1\)- Second term: \(\binom{3}{1}(1)^2(a^3)^1 = 3a^3\)- Third term: \(\binom{3}{2}(1)^1(a^3)^2 = 3a^6\)- Fourth term: \(\binom{3}{3}(1)^0(a^3)^3 = a^9\)

Step 5: Combine the Expanded Terms

Combine all the terms found in Step 4:\[1 + 3a^3 + 3a^6 + a^9\]

Key Concepts

Polynomial Expansion

Polynomial expansion is a crucial concept in algebra which includes expanding expressions that are raised to a power, particularly binomials, in order to simplify or solve problems. In the expression \((1+a^3)^3\), the task is to expand the binomial. This involves transforming a compact expression into a longer polynomial.
To perform polynomial expansion, especially with binomials, the Binomial Theorem is frequently used. This theorem allows us to expand a binomial expression \((x+y)^n\) into a sum involving binomial coefficients.
By applying the theorem, we generate all the terms of the expanded polynomial, each of which includes powers of both components in the binomial. This demonstrates how an expression can be rewritten in a more detailed form characterized by sums of various powers.
Understanding polynomial expansion equips students with the ability to manage complex algebraic expressions by breaking them down into simpler, more workable parts.

Binomial Coefficients

The concept of binomial coefficients is essential when expanding binomials using the Binomial Theorem. These coefficients, denoted as \(\binom{n}{k}\), serve as multipliers for each term in the expanded expression.

• In our problem, where \((1+a^3)^3\), the binomial coefficients are obtained from the expansion of \((x+y)^n\) using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• Here, they are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).
• These values represent the different ways in which terms can combine when raised to a power.

Understanding binomial coefficients allows us to know the relative weights each term carries in the final expanded expression. This is vital for arranging and simplifying expressions efficiently. By appreciating the role of these coefficients, students can better comprehend distribution and combination principles within algebraic operations.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form, making them easier to understand and work with. After expanding a binomial such as \((1+a^3)^3\), it's necessary to simplify the result: \(1 + 3a^3 + 3a^6 + a^9\).
This process involves combining like terms and organizing the expression in a way that highlights its structure. Though in this particular example, each term is inherently unique, other expressions may require combing similar terms to reduce to a simpler form.

• Simplification makes expressions less cumbersome and more intuitive.
• Properly simplified results are essential when solving algebraic equations, as they reveal the expression's key characteristics.

Mastering algebraic simplification promotes problem-solving efficiency and aids in grasping more advanced concepts, enhancing overall mathematical literacy.