Problem 36 - College Algebra

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{17} $$

Step-by-Step Solution

Answer

(x^{2}+1)^{17} equates to \(x^{34} + 17x^{32} + 136x^{30} + ...\)

Step 1: Apply Binomial Theorem

Using the Binomial Theorem, the first three terms of the expansion of \((x^{2}+1)^{17}\) are given by: \[ (x^{2}+1)^{17} = \binom{17}{0} (x^{2})^{17} (1)^{0} + \binom{17}{1} (x^{2})^{16} (1)^{1} + \binom{17}{2} (x^{2})^{15} (1)^{2} + ...\]

Step 2: Calculate terms

Calculating the first three terms: \[ \text{First term} = \binom{17}{0} (x^{2})^{17} (1)^{0} = x^{34}\] \[ \text{Second term} = \binom{17}{1} (x^{2})^{16} (1)^{1} = 17x^{32}\] \[ \text{Third term} = \binom{17}{2} (x^{2})^{15} (1)^{2} = 136x^{30}\]

Step 3: Write the full simplified expression

The full simplified three terms of the given binomial expression is: \[ (x^{2}+1)^{17} = x^{34} + 17x^{32} + 136x^{30} + ... \]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful method in algebra that simplifies the expansion of expressions raised to a high power. It allows for the expansion of any binomial raised to a positive integer power. In its essence, the theorem provides a formula to expand expressions of the form

• \((a+b)^n\).

This means when you have a binomial—an expression with two terms—and you raise it to a power \(n\), the Binomial Theorem helps us find each term in the expansion.Each component of the theorem involves binomial coefficients, which tell us how many ways we can select items from a larger set, along with our variables raised to appropriate powers. For instance,

• expression \((x^2 + 1)^{17}\) is expanded using the theorem.

The first part of expanding based on this theorem is identifying the number of terms. Then, we use the binomial coefficients (often referred to as Pascal's Triangle) to determine how each term's coefficient is calculated.

Binomial Coefficients

Binomial coefficients are essential when using the Binomial Theorem. They are denoted by

• \(\binom{n}{k}\)

and represent the number of ways to choose \(k\) items from \(n\) without regard to the order of selection.These coefficients can be found easily using the formula:

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

Where \(!\) stands for factorial, meaning the product of all positive integers up to that number.In the problem \((x^2 + 1)^{17}\), the first three coefficients were calculated:

• For \(k=0\), \(\binom{17}{0} = 1\)
• For \(k=1\), \(\binom{17}{1} = 17\)
• For \(k=2\), \(\binom{17}{2} = 136\)

These coefficients dramatically affect the size of each term, showcasing patterns and symmetry often referred to in Pascal's Triangle.

Polynomial Expressions

Polynomial expressions consist of terms that are combinations of variables raised to non-negative integer powers, multiplied by coefficients. These expressions take a form such as \(ax^n + bx^{n-1} + cx^{n-2} + \ldots + zx^0\), where \(a, b, c, ..., z\) are constants.In binomial expansion, each term in the polynomial results from raising each part of the binomial to the respective powers and combining with binomial coefficients. The power and structure of polynomial expressions determine the behavior of functions and solutions.In the expansion of \((x^2 + 1)^{17}\):

• The terms were expanded to get powers like \(x^{34}, x^{32}, x^{30}\).
• The exponents here indicate how each term of the polynomial expression is multiplied, reflecting characteristics such as symmetry and degree.

Understanding polynomial expressions is crucial for accurately predicting how changing variables can influence the overall expression and solutions in algebra.