Problem 50 - College Algebra

Question

In Exercises 49-52, use the Binomial Theorem to expand each expression and write the result in simplified form. $$\left(x^{2}+x^{-3}\right)^{4}$$

Step-by-Step Solution

Answer

The expanded and simplified form of the expression \((x^{2}+x^{-3})^{4}\) is \(x^8 + 4x^3 + 6x^{-2} + 4x^{-7} + x^{-12}\).

Step 1: Identify the values of a, b, and n

In this case, we see that \(a = x^2\), \(b = x^{-3}\), and \(n = 4\).

Step 2: Use Binomial Theorem to expand the expression

According to the Binomial Theorem, we expand the expression as following: \((x^{2}+x^{-3})^{4} = \sum_{k=0}^{4} {4\choose k}x^{2(4-k)}(x^{-3})^{k}\). Based on this, there are 5 terms in the result.

Step 3: Calculate each term

Write down every term while calculating binomial coefficients: \({4\choose 0}x^{2(4-0)}(x^{-3})^0 = 1*(x^8)*1 = x^8\), \({4\choose 1}x^{2(4-1)}(x^{-3})^1 = 4*(x^6)*x^{-3} = 4x^3\), \({4\choose 2}x^{2(4-2)}(x^{-3})^2 = 6*(x^4)*(x^{-6}) = 6x^{-2}\), \({4\choose 3}x^{2(4-3)}(x^{-3})^3 = 4*(x^2)*(x^{-9}) = 4x^{-7}\), \({4\choose 4}x^{2(4-4)}(x^{-3})^4 = 1*1*(x^{-12}) = x^{-12}\).

Step 4: State the final result

Combine all terms to write out the final result: \(x^8 + 4x^3 + 6x^{-2} + 4x^{-7} + x^{-12}\).

Key Concepts

Simplifying Algebraic Expressions

In mathematics, simplifying algebraic expressions is an essential skill, allowing us to present equations and formulas in their most concise and understandable form. This involves combining like terms, which are terms that contain the same variables raised to the same power, and performing operations like addition and subtraction accordingly.

For instance, in our given exercise \( (x^{2}+x^{-3})^{4} \), we first expand the expression using the Binomial Theorem, which results in a series of terms that include powers of \( x \). These terms might seem complex at first, but upon closer examination, we notice there are no like terms to combine, as each term contains a unique power of \( x \). As a result, the simplified form of the expression is the same as the expanded form, showing each term clearly separated by plus signs. Understanding this process is crucial for tackling algebra with confidence and ensures that complex polynomial expressions become more manageable.

Binomial Coefficients

Binomial coefficients are the numbers that feature in the binomial expansion and are represented symbolically as \( {n\choose k} \). These coefficients correspond to the elements of Pascal's Triangle and determine how many different ways \( k \)-sized subsets can be chosen from a larger set of \( n \)-items. In the context of binomial expansion, they give us the coefficients for each term in the expanded expression.

With our exercise \( (x^{2}+x^{-3})^{4} \), each term in the expansion \( {4\choose k}x^{2(4-k)}(x^{-3})^{k} \) includes the binomial coefficient \( {4\choose k} \), starting from \( k = 0 \) up to \( k = 4 \). Calculating each one, we get 1, 4, 6, 4, and 1 as the coefficients for the corresponding terms. These coefficients are crucial as they determine the weight of each term in the polynomial, reflecting its combinatorial significance in the expansion process.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial, which is raised to a power, as a series of terms added together without any exponents. The Binomial Theorem is a formula that provides a direct method for expanding expressions of the form \( (a+b)^n \) where \( n \) is a non-negative integer.

In the series expansion of the expresssion \( (x^{2}+x^{-3})^{4} \) provided by the Binomial Theorem, each term will be a product of a binomial coefficient, a power of the first term \( a \) (in this case, \( x^2 \)), and a power of the second term \( b \) (in this case, \( x^{-3} \) ). Each term's exponents, when added together, must sum to \( n \) (which is 4 in our example). After the expansion, we are left with the polynomial in its extended form, showcasing the power of the Binomial Theorem in simplifying the process of polynomial expansion significantly. It is paramount for students to grasp this technique as it is widely used in different areas of algebra and calculus.