Problem 15 - College Algebra

Question

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 the given binomial expression \((2x+1)^{4}\) is \(16x^4 + 32x^3 + 24x^2 + 8x + 1\).

Step 1: Express the Binomial

The expression in the problem is \((2x+1)^{4}\). It's noted that the binomial theorem for \( (a + b)^n \) is given by \( \sum_{i=0}^{n} {n \choose i} a^{n-i}b^{i} \). where \({n \choose i}\) is a binomial coefficient.

Step 2: Apply the Binomial Theorem

We can replace 'a' with '2x', 'b' with '1', and 'n' with '4' in the given binomial theorem. It would look like this: \((2x+1)^4 = \sum_{i=0}^{4} {4 \choose i} (2x)^{4-i} 1^{i} \).

Step 3: Simplify

Let's simplify this sum: \(16x^4 + {4 \choose 1} * 8x^3 * 1 + {4 \choose 2} * 4x^2 * 1 + {4 \choose 3} * 2x * 1 + 1 = 16x^4 + 32x^3 + 24x^2 + 8x + 1.\)

Key Concepts

Binomial Expansion

The binomial theorem provides a concise way to expand expressions raised to a power. This is particularly useful for binomials like \((2x+1)^4\). The theorem states that the expansion of \((a+b)^n\) can be expressed as the sum of terms involving powers of \(a\) and \(b\), multiplied by specific coefficients called binomial coefficients.
To illustrate, in any given expansion derived from the theorem, the term structure takes the form of \( a^{n-i} b^i \), where \(i\) varies from 0 to \(n\). Each of these terms also includes a binomial coefficient, calculated based on the position \(i\) within the sum.
Applying this method to \((2x+1)^4\), you'll end up with an expression composed of five terms, each corresponding to different powers of \(2x\) and 1.

Binomial Coefficients

Binomial coefficients play an integral part in the expansion process. These coefficients are denoted as \({n \choose i}\), where \(n\) is the exponent and \(i\) is the specific term position within the expansion.
They quantify the number of ways to choose \(i\) successes out of \(n\) trials in a binomial probability distribution, which ties their combinatorial origins to algebra.

• For example, to find the coefficient of the third term in the expansion of \((2x+1)^4\), calculate \({4 \choose 2}\).

The actual value of \({n \choose i}\) can be computed using the formula \(\frac{n!}{i!(n-i)!}\). This formula highlights its ties to permutations and combinations. With these coefficients, each term in the binomial expansion reflects contributions from both sequences of powers and arrangements of elements.

Algebraic Expressions

Algebraic expressions represent mathematical phrases that can involve numbers, variables, and operators like addition and multiplication.
In the context of binomial expansions, these expressions denote each individual term within the equation once expanded.

• For instance, when you expand \((2x+1)^4\), each term is an algebraic expression such as \(16x^4\), \(32x^3\), and so forth.
• When simplified, the expansion of \((2x+1)^4\) provides a set of distinct algebraic expressions.

Simplification is a key step, which ensures that the expanded form is easier to interpret and manage.
By applying arithmetic calculations, such as combining like terms, the result becomes more concise and effective for further mathematical manipulation or analysis. This results in a polynomial expression such as \(16x^4 + 32x^3 + 24x^2 + 8x + 1\).