Problem 45 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$\left(x^{2}+1\right)^{3}$$

Step-by-Step Solution

Answer

Using the binomial theorem, the expanded expression for \(\left(x^{2}+1\right)^{3}\) is \(x^6 + 3x^4 + 3x^2 + 1\).

Step 1: Identify Variables

In the given expression \(\left(x^{2}+1\right)^{3}\), identify the values for a, b, and n: - \(a = x^2\) - \(b = 1\) - \(n = 3\)

Step 2: Calculate Binomial Coefficients

Using the formula for binomial coefficients \(\binom{n}{k}\), calculate the coefficients for each term in the expansion, where \(n = 3\): - \(\binom{3}{0} = 1\) - \(\binom{3}{1} = 3\) - \(\binom{3}{2} = 3\) - \(\binom{3}{3} = 1\)

Step 3: Calculate Terms of the Expansion

Using the binomial theorem and the calculated coefficients, compute the terms of the expansion: Term 1: - Coefficient: \(\binom{3}{0} = 1\) - \((x^2)^{3-0}(1)^{0} = x^6\) - Term 1: \(1x^6\) Term 2: - Coefficient: \(\binom{3}{1} = 3\) - \((x^2)^{3-1}(1)^{1} = x^4\) - Term 2: \(3x^4\) Term 3: - Coefficient: \(\binom{3}{2} = 3\) - \((x^2)^{3-2}(1)^{2} = x^2\) - Term 3: \(3x^2\) Term 4: - Coefficient: \(\binom{3}{3} = 1\) - \((x^2)^{3-3}(1)^{3} = 1\) - Term 4: \(1\)

Step 4: Combine Terms

Finally, combine the terms obtained in Step 3 to get the expanded expression: \[\left(x^{2}+1\right)^{3} = 1x^6 + 3x^4 + 3x^2 + 1\]

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to represent numbers and relationships abstractly. In algebra, letters such as \(x\) or \(a\) are used to stand for numbers. This provides a way to solve problems in a general form.

The expression \((x^2 + 1)^3\) involves algebraic structures and requires the application of mathematical rules to manipulate and simplify. Mastery of algebra is essential for understanding how to expand and simplify expressions, solve equations, and explore relationships between variables.

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. Each term in a polynomial is made up of a coefficient and a variable raised to a non-negative integer power.

• For example, in the expression \(x^6 + 3x^4 + 3x^2 + 1\), each term like \(3x^4\) is a part of the polynomial.
• Polynomials can be simple constants, like \(1\), or more complex, involving multiple terms with various degrees.

Understanding polynomials is important for expanding expressions like \((x^2 + 1)^3\), where knowledge of how to handle different terms and coefficients is key to solving the problem.

Binomial Expansion

Binomial expansion is the process of expanding powers of a binomial expression using the binomial theorem. The theorem provides a formula to expand expressions like \((a + b)^n\) into a series of terms.

Given \((x^2 + 1)^3\), the binomial theorem allows us to express it as \(1x^6 + 3x^4 + 3x^2 + 1\). Each term of the expansion involves a binomial coefficient, and the powers of \(a\) and \(b\) decrease and increase, respectively, through the expansion.
Binomial expansion simplifies problems involving higher powers by breaking them down into manageable parts.

Binomial Coefficients

Binomial coefficients are numerical factors that multiply each term in a binomial expansion. They are represented as \(\binom{n}{k}\) and calculate the number of ways to choose \(k\) elements from a set of \(n\) elements.

For the example \((x^2 + 1)^3\):

• The coefficients are \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).

These coefficients guide the number of ways each term can be formed and are central to the application of the binomial theorem.
Understanding these coefficients helps in predicting and calculating the expanded form of binomial expressions efficiently.