Problem 15 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(2 x+1)^{4}$$

Step-by-Step Solution

Answer

The simplified form of \((2x+1)^4\) is \(16x^{4}+32x^{3}+24x^{2}+8x+1\).

Step 1: Identify the Form of Binomial Theorem

The Binomial Theorem can be represented as follows: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]\where \( \binom{n}{k} \) is the binomial coefficient which can be computed using factorials: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]. So, the same would be applied here using \((a+b)^n = (2x+1)^4\)

Step 2: Apply the Binomial Theorem

Now, by substituting \( a = 2x \), \( b = 1 \) and \( n = 4 \) into the formula and expanding, we have\ \[ (2x+1)^4=\binom{4}{0}(2x)^{4}+ \binom{4}{1}(2x)^{3}(1)+ \binom{4}{2}(2x)^{2}(1)^{2}+ \binom{4}{3}(2x)(1)^{3}+ \binom{4}{4}(1)^{4} \]

Step 3: Compute the Binomial Coefficients and Simplify

Solving for each term separately \ \[ \binom{4}{0}(2x)^{4}=16x^{4} \]\ \[ \binom{4}{1}(2x)^{3}(1)=32x^{3} \]\ \[ \binom{4}{2}(2x)^{2}(1)^{2}=24x^{2} \]\ \[ \binom{4}{3}(2x)(1)^{3}=8x \]\ \[ \binom{4}{4}(1)^{4}=1 \]\ Thus, the simplified form of the binomial expression \((2x+1)^4\) is :\ \[ (2x+1)^4=16x^{4}+32x^{3}+24x^{2}+8x+1 \]

Key Concepts

Binomial Expansion

The binomial expansion is a way of expanding expressions that are raised to a power. For example, when you have an expression like \((2x + 1)^4\), you can expand it using the binomial theorem. This theorem allows you to express the power as a sum of terms involving binomial coefficients, powers of the first term, and powers of the second term.

To use the binomial theorem, identify your binomial expression \((a + b)^n\), where:

• \(a\) is the first term (in our example, \(2x\))
• \(b\) is the second term (in our example, \(1\))
• \(n\) is the power to which the binomial is raised (in our example, \(4\))

The expansion follows this pattern, using a combination of binomial coefficients and decreasing powers of \(a\) while increasing the powers of \(b\).

The expansion for \((2x + 1)^4\) produces several terms, each involving a binomial coefficient, resulting in a polynomial like \(16x^4 + 32x^3 + 24x^2 + 8x + 1\). Each part of this expansion is a systematic application of the binomial theorem.

Binomial Coefficient

The binomial coefficient is a key element in the binomial expansion. Notated as \(\binom{n}{k}\), it represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order.

This coefficient can be found using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here, \(n!\) (pronounced "n factorial") is the product of all positive integers up to \(n\). It calculates how many different ways you can arrange the items in a set.

For example, in the expression \((2x + 1)^4\), we calculate binomial coefficients like \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), \(\binom{4}{3}\), and \(\binom{4}{4}\), leading to values such as 1, 4, 6, 4, and 1 respectively. These coefficients are used to scale the terms in the expansion.

Factorials

Factorials are a fundamental mathematical concept critical in computing binomial coefficients. A factorial of a non-negative integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). It is defined as follows:

• \(0! = 1\)
• \(1! = 1\)
• \(2! = 2\times 1 = 2\)
• \(3! = 3\times 2\times 1 = 6\)
• \(4! = 4\times 3\times 2\times 1 = 24\)

Factorials grow very quickly as the number \(n\) increases. They help us determine the binomial coefficients used in binomial expansions.

In our problem, to calculate coefficients like \(\binom{4}{2}\), you'd compute \(\frac{4!}{2!(4-2)!}\), resulting in \(\frac{24}{2 \times 2} = 6\). Understanding how to use factorials is crucial for finding out these coefficients and successfully expanding binomials.