Problem 10 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x+4)^{3}$$

Step-by-Step Solution

Answer

The expanded form of \((x+4)^{3}\) is \(x^{3} + 12x^{2} + 48x + 64\).

Step 1: Understand the Binomial Theorem

The Binomial theorem states that \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \), where \( \binom{n}{k} \) is the binomial coefficient 'n choose k'. The theorem describes the expanded form of the binomial \( (a+b)^n \) in terms of \( a \) and \( b \), with coefficients specific to \( n \).

Step 2: Apply the Binomial Theorem to the expression

Apply the Binomial theorem in this specific case with \( n = 3 \), \( a = x \) and \( b = 4 \): \( (x+4)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k}4^k = \binom{3}{0} x^{3}4^0 + \binom{3}{1} x^{2}4^1 + \binom{3}{2} x^{1}4^2 + \binom{3}{3} x^{0}4^3 \)

Step 3: Calculate the binomial coefficients and simplify

Calculate the binomial coefficients, reduce and simplify each term: \\ \( \binom{3}{0} x^{3}4^{0} = 1*x^{3}*1 = x^{3} \) \\ \( \binom{3}{1} x^{2}4^{1} = 3*x^{2}*4 = 12x^{2} \) \\ \( \binom{3}{2}x^{1}4^{2} = 3*x*16 = 48x \) \\ \( \binom{3}{3}x^{0}4^{3} = 1*1*64 = 64 \) \\ The expanded form of \((x+4)^{3}\) is \(x^{3} + 12x^{2} + 48x + 64\).

Key Concepts

Binomial Expansion

The binomial expansion is an essential concept in algebra that allows us to expand expressions of the form \((a+b)^n\). Instead of multiplying the binomial by itself repeatedly, which can be tedious, the Binomial Theorem provides a straightforward method to find all the terms in the expansion.

The formula for the Binomial Theorem is:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\)

Here, \(n\) represents the power to which the binomial is raised. The symbol \(\binom{n}{k}\) is crucial as it helps in calculating the coefficients of each term. These expansions are useful in various areas of mathematics, including probability, statistics, and algebra. Understanding how to apply this theorem helps simplify and solve complex polynomial equations effectively.

Binomial Coefficients

Binomial coefficients are integral to the binomial expansion process. A binomial coefficient is represented by \(\binom{n}{k}\), also known as 'n choose k'.

This coefficient determines how many ways you can choose \(k\) successes from \(n\) trials, which in terms of binomial expansion, gives the coefficient of each term.

• The formula for calculating a binomial coefficient is: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Here, \(!\) denotes factorial, which is the product of all positive integers up to that number. For example, \(3! = 3 \times 2 \times 1 = 6\).

By calculating binomial coefficients, you determine the weighting of each term in the expansion. These coefficients play an essential role in simplifying polynomial expressions, as seen in the binomial expansion of \((x+4)^3\).

Polynomial Expansion

Polynomial expansion involves expressing a polynomial in its long-form, where each term is expanded and simplified. The binomial expansion, in particular, is a type of polynomial expansion specific to binomials raised to a power.

For example, when expanding \((x+4)^3\) using the Binomial Theorem, you identify each term by combining the different powers of \(x\) and \(4\) from the binomial. When expanded, this becomes:

• \(x^3\)
• \(+ 12x^2\)
• \(+ 48x\)
• \(+ 64\)

Each term results from the product of a power of \(x\) and \(4\), multiplied by its respective binomial coefficient.

The process of polynomial expansion transforms a compact expression into a series of terms, each clearly defined, which can then be analyzed or further simplified as required. This technique is fundamental to understanding the behavior of polynomials and honing skills in algebra.