Problem 30 - Algebra and Trigonometry

Question

Find the first four terms in the expansion of \(\left(x^{1 / 2}+1\right)^{30}\)

Step-by-Step Solution

Answer

The first four terms are: \(x^{15} + 30x^{14.5} + 435x^{14} + 4060x^{13.5}\).

Step 1: Understand the Binomial Theorem

The binomial theorem allows us to expand expressions of the form \((a+b)^n\). It states that the expansion of \((a+b)^n\) is given by the sum: \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(\binom{n}{k}\) represents the binomial coefficient, which can be calculated as \(\binom{n}{k} = \frac{n!}{k! (n-k)!}\).

Step 2: Identify Variables in the Expression

For the expression \((x^{1/2}+1)^{30}\), we have the variables:- \(a = x^{1/2}\)- \(b = 1\)- \(n = 30\)

Step 3: Calculate the First Term

Using the binomial theorem formula, the first term is obtained with \(k=0\):\[ T_0 = \binom{30}{0} (x^{1/2})^{30-0} 1^0 = 1 \times x^{15} = x^{15} \]

Step 4: Calculate the Second Term

For the second term, we use \(k=1\):\[ T_1 = \binom{30}{1} (x^{1/2})^{30-1} 1^1 = 30 \times x^{14.5} = 30 x^{14.5} \]

Step 5: Calculate the Third Term

For the third term, use \(k=2\):\[ T_2 = \binom{30}{2} (x^{1/2})^{30-2} 1^2 = \frac{30 \times 29}{2} \times x^{14} = 435 x^{14} \]

Step 6: Calculate the Fourth Term

For the fourth term, use \(k=3\):\[ T_3 = \binom{30}{3} (x^{1/2})^{30-3} 1^3 = \frac{30 \times 29 \times 28}{6} \times x^{13.5} = 4060 x^{13.5} \]

Step 7: Combine the First Four Terms

Combine all terms calculated: \(T_0 + T_1 + T_2 + T_3 = x^{15} + 30 x^{14.5} + 435 x^{14} + 4060 x^{13.5}\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful technique used in algebra to expand expressions of the form \((a+b)^n\). This method allows us to express a power of a binomial as a sum of terms involving binomial coefficients and the variables raised to varying powers. For example, in the expression \((x^{1/2} + 1)^{30}\), the binomial expansion enables us to systematically determine each term in the expansion according to the formula defined by the binomial theorem. In general, the expansion takes the shape of:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• Where \(\binom{n}{k}\) is the binomial coefficient, calculated as \(\frac{n!}{k! (n-k)!}\)

By using the binomial expansion, you can simplify complex expressions and make calculations more manageable. It lays the foundation for exploring polynomials and other algebraic structures, enhancing both theoretical understanding and practical problem-solving skills.

Binomial Coefficient

A binomial coefficient is a crucial component of binomial expansion, representing the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection. The notation \(\binom{n}{k}\) indicates a binomial coefficient, which can be computed using the formula:

• \(\binom{n}{k} = \frac{n!}{k! (n-k)!}\)

Understanding how to calculate binomial coefficients enables us to determine each term's contribution in a binomial expansion without performing repetitive calculations. For example, in expanding \((x^{1/2} + 1)^{30}\), we encountered coefficients such as \(\binom{30}{0} = 1\), \(\binom{30}{1} = 30\), \(\binom{30}{2} = 435\), and \(\binom{30}{3} = 4060\). Each coefficient corresponds to a specific term's contribution in the expansion, providing critical numerical factors that multiply the variable components.

Polynomial Expressions

A polynomial expression is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the context of the binomial theorem, polynomial expressions emerge when expanding binomials. The resulting terms follow a structured pattern:

• Variable components have gradually decreasing powers for one variable and increasing powers for the other.
• The coefficients come from binomial coefficients, indicating each term's weight or significance.

Understanding polynomial expressions helps in recognizing familiar patterns in algebra and calculus. For instance, in our example \((x^{1/2} + 1)^{30}\), each of the first four terms—\(x^{15} + 30x^{14.5} + 435x^{14} + 4060x^{13.5}\)—is an individual polynomial expression that contributes to the larger polynomial formed by the binomial expansion. Such expressions are instrumental in solving equations, analyzing functions, and modeling various phenomena in mathematics and related fields.