Problem 6 - Algebra and Trigonometry

Question

Use Pascal’s triangle to expand the expression. $$ (2 x+1)^{4} $$

Step-by-Step Solution

Answer

\((2x+1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1\)

Step 1: Understand the problem

We need to use Pascal's triangle to expand the expression \((2x + 1)^4\). Pascal's triangle provides the coefficients for the binomial expansion \((a + b)^n\).

Step 2: Identify the correct row in Pascal's triangle

For an expansion of \((a + b)^n\), the coefficients are found in the \(n\)-th row of Pascal's triangle. For \(n = 4\), the row is \([1, 4, 6, 4, 1]\).

Step 3: Write the general term of the binomial expansion

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 2x\), \(b = 1\), and \(n = 4\).

Step 4: Apply the coefficients to compute each term

We apply the coefficients from Pascal's triangle to the formula. The terms are:\- First term: \(1 \, (2x)^4(1)^0 = 16x^4\)\- Second term: \(4 \, (2x)^3 (1)^1 = 32x^3\)\- Third term: \(6 \, (2x)^2 (1)^2 = 24x^2\)\- Fourth term: \(4 \, (2x)^1 (1)^3 = 8x\)\- Fifth term: \(1 \, (2x)^0 (1)^4 = 1\)

Step 5: Write the expanded form

Combine all the calculated terms to get the expanded form: \(16x^4 + 32x^3 + 24x^2 + 8x + 1\).

Key Concepts

Binomial Expansion

Binomial expansion is a fundamental concept in algebra that involves expanding expressions of the form \((a + b)^n\) into a sum of terms involving powers of \(a\) and \(b\). This process transforms a simple expression into a polynomial. For example, the expression \((2x + 1)^4\) can be expanded using binomial expansion to reveal a detailed polynomial with several terms. Each term in this expansion is a product of a numerical coefficient, a power of \(a\), and a power of \(b\). Binomial expansions are useful for simplifying complex algebraic expressions and can be visualized using Pascal's triangle, which provides the coefficients needed for the expansion.

Binomial Theorem

The binomial theorem is a powerful mathematical tool that provides a formula for expanding any binomial expression \((a + b)^n\) into a sum of terms. This theorem is central to understanding how binomial expansion works. The formula given by the binomial theorem is: \[ (a + b)^n = \sum_{k=0}^{n}\binom{n}{k} a^{n-k} b^k \] Here, the symbol \(\binom{n}{k}\) represents the binomial coefficients, which can be directly retrieved from Pascal’s triangle. These coefficients dictate the multiplicative factor for each term in the expansion. The binomial theorem not only simplifies the process of expansion but also reveals the pattern and symmetry within polynomial expressions.

Polynomial Expansion

Polynomial expansion refers to expressing a power of a binomial as a sum of individual terms, each involving the variables with non-negative integer powers. When we apply the binomial theorem to an expression such as \((2x + 1)^4\), the polynomial expansion is achieved by taking each term sequentially: - The coefficients derived from Pascal’s triangle: \([1, 4, 6, 4, 1]\) aid in forming the terms. - Applying these coefficients provides us terms like \(16x^4\), \(32x^3\), and so on. - Each term results from a consistent method involving the multiplication of a coefficient, a power of \(a\) and a power of \(b\). Ultimately, the result of polynomial expansion simplifies binomial expressions and facilitates further algebraic manipulation.

Algebra

Algebra is a vast field of mathematics that includes the study of mathematical symbols and the rules for manipulating these symbols. It allows us to solve equations, understand polynomial structures, and delve deeper into various mathematical concepts, such as the binomial theorem. In the context of binomial and polynomial expansions, algebra serves as the foundational language and framework through which these expansions are understood. Algebraic skills enable you to:

• Simplify complex expressions.
• Manipulate equations to find solutions.
• Utilize formulas like the binomial theorem for expanding expressions.

These algebraic techniques are not just abstract ideas but practical tools that enhance one's ability to address diverse mathematical challenges.