Problem 16 - Algebra and Trigonometry

Question

Use Pascal’s triangle to expand the expression. $$ \left(2+\frac{x}{2}\right)^{5} $$

Step-by-Step Solution

Answer

The expansion is \(32 + 40x + 20x^2 + 5x^3 + \frac{5x^4}{8} + \frac{x^5}{32}\).

Step 1: Identify the Expansion Coefficients

To expand the expression \((2+\frac{x}{2})^5\), we first identify the coefficients in Pascal’s triangle for the binomial expansion with the exponent \(n=5\). The row corresponding to \(n=5\) is \(1, 5, 10, 10, 5, 1\). These are the coefficients we will use in the expansion.

Step 2: Write the General Binomial Expansion Formula

The binomial theorem states that for an expression of the form \((a+b)^n\), the expansion is \(a^nC_0 + a^{n-1}bC_1 + \ldots + b^nC_n\), where \(C_k\) are the coefficients from Pascal's triangle. For \((2+\frac{x}{2})^5\), \(a=2\), \(b=\frac{x}{2}\), and we will use the corresponding coefficients: \(1, 5, 10, 10, 5, 1\).

Step 3: Calculate Each Term in the Expansion

We calculate each term using the form \(C_k \cdot a^{n-k} \cdot b^k\):- Term 1: \(1 \cdot (2)^5 \cdot (\frac{x}{2})^0 = 32\)- Term 2: \(5 \cdot (2)^4 \cdot (\frac{x}{2})^1 = 5 \cdot 16 \cdot \frac{x}{2} = 40x\)- Term 3: \(10 \cdot (2)^3 \cdot (\frac{x}{2})^2 = 10 \cdot 8 \cdot \frac{x^2}{4} = 20x^2\)- Term 4: \(10 \cdot (2)^2 \cdot (\frac{x}{2})^3 = 10 \cdot 4 \cdot \frac{x^3}{8} = 5x^3\)- Term 5: \(5 \cdot (2)^1 \cdot (\frac{x}{2})^4 = 5 \cdot 2 \cdot \frac{x^4}{16} = \frac{5x^4}{8}\)- Term 6: \(1 \cdot (2)^0 \cdot (\frac{x}{2})^5 = \frac{x^5}{32}\)

Step 4: Combine the Terms to Form the Binomial Expansion

The expanded expression is obtained by adding all the terms calculated in the previous step:\[ (2 + \frac{x}{2})^5 = 32 + 40x + 20x^2 + 5x^3 + \frac{5x^4}{8} + \frac{x^5}{32} \]

Key Concepts

Pascal's Triangle

Pascal's Triangle is a powerful tool for finding coefficients in binomial expansions. It is a triangular array of numbers, where each number is the sum of the two numbers directly above it. This structure can be built row by row, starting with a simple "1" at the top.

When expanding expressions like \((a + b)^n\), you find the row in Pascal's Triangle that corresponds to \(n\). The numbers in this row are the coefficients for each term in the binomial expansion. For example, the row for \(n = 5\) is \(1, 5, 10, 10, 5, 1\).

This technique is especially helpful in simplifying algebraic operations since it provides a quick way to find coefficients without having to manually multiply out each product.

Binomial Theorem

The Binomial Theorem is a formula used for expanding expressions raised to a power, such as \((a + b)^n\). It states that the expansion can be expressed as a series:

• \(a^nC_0 + a^{n-1}bC_1 + \cdots + b^nC_n\)

This means each term consists of

• \(a\) raised to a decreasing power starting from \(n\),
• multiplied by \(b\) raised to an increasing power starting from zero,
• and each term is multiplied by a coefficient \(C_k\) from Pascal's Triangle.

To apply the theorem, simply identify \(a\) and \(b\) in your binomial, choose your \(n\), then use the corresponding row of coefficients from Pascal's Triangle. This method makes binomial expansion systematic and efficient.

Expansion Coefficients

Expansion coefficients are the numbers that multiply each term in a binomial expansion. When using Pascal's Triangle or the Binomial Theorem, these coefficients are crucial for determining the terms of the expanded form.

Let's say you have an expression like \((a + b)^5\). Using Pascal's Triangle, the coefficients you'd use are \(1, 5, 10, 10, 5, 1\). This means the first term of your expansion would be multiplied by 1, the second by 5, the third and fourth by 10, the fifth by 5, and the last by 1.

These coefficients ensure each term is correctly balanced in the equation and are essential for accurately expanding any binomial expression. They are found easily using Pascal's Triangle or combinations formulas, making the expansion process smoother and more manageable.

Polynomial Expressions

Polynomial expressions are sums of terms involving variables raised to non-negative integer powers. When dealing with binomial expansions, the result you get is a polynomial. For instance, expanding \((2 + \frac{x}{2})^5\) yields a polynomial:

• \(32\)
• \(+ 40x\)
• \(+ 20x^2\)
• \(+ 5x^3\)
• \(+ \frac{5x^4}{8}\)
• \(+ \frac{x^5}{32}\)

The power of the polynomial is determined by the power to which the binomial is raised, in this case, 5. Each term in the resulting polynomial reflects the coefficients from Pascal’s Triangle and powers of \(a\) and \(b\) arranged according to the Binomial Theorem. Understanding polynomials through binomial expansions allows for a deeper comprehension of their structure and behavior.