Problem 19 - Contemporary Precalculus

Question

Expand and (where possible) simplify the expression. $$(x+1)^{4}$$

Step-by-Step Solution

Answer

Answer: The expanded and simplified form of the expression \((x+1)^4\) is \(x^4 + 4x^3 + 6x^2 + 4x + 1\).

Step 1: Apply the Binomial Theorem

Using the binomial theorem, we can expand \((x+1)^4\) as follows: \((x+1)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} 1^k\)

Step 2: Evaluate the Binomial Coefficients

Now, we will compute the binomial coefficients for the given expression: \(\binom{4}{0} = 1\) \(\binom{4}{1} = 4\) \(\binom{4}{2} = 6\) \(\binom{4}{3} = 4\) \(\binom{4}{4} = 1\)

Step 3: Expand the Expression

With the binomial coefficients, we can now expand \((x+1)^4\): \((x+1)^4 = \binom{4}{0} x^{4} 1^0 + \binom{4}{1} x^{3} 1^1 + \binom{4}{2} x^{2} 1^2 + \binom{4}{3} x^1 1^3 + \binom{4}{4} x^0 1^4\)

Step 4: Simplify the Expression

We can replace the binomial coefficients and simplify the expression: \((x+1)^4 = 1x^4 + 4x^3 + 6x^2 + 4x^1 + 1\) Thus, the expanded and simplified expression is: $$(x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1$$

Key Concepts

Polynomial Expansion

Polynomial expansion is a technique used to express a raised binomial of the form \((x + a)^n\) in its expanded form. This process involves breaking down the binomial to reveal each term in the polynomial sequence. Let's explore this concept using \((x+1)^4\) as an example.

To start, the binomial theorem provides a framework for this type of expansion. The theorem states that \((x + a)^n\) can be expanded into a sum of terms \(\sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k\). In our case, this means that the binomial products are distributed and summed up into individual terms.

This powerful tool helps to convert a compact binomial expression into a detailed polynomial with clear, individual components. Each coefficient of the expanded polynomial corresponds to a special number called a binomial coefficient.

Binomial Coefficients

Binomial coefficients are central to the process of expanding binomials. They are the numbers that appear as coefficients in the polynomial expansion and are denoted using the notation \(\binom{n}{k}\). These coefficients can be easily found using Pascal's Triangle or calculated directly through the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

In the specific example of \((x+1)^4\), the binomial coefficients are obtained as follows:

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

These coefficients are then used to scale each term in the expansion of the binomial. Understanding them is key because they determine how much influence each power of \(x\) and \(1\) will have in the polynomial.

These numbers create a symmetrical pattern, highlighting a nice property of binomial expansions where the coefficients reflect a pattern of increasing to a peak and then decreasing.

Simplifying Expressions

Once a binomial is expanded using the binomial theorem, the next critical step is to simplify the expression. Simplifying helps transform the complex bins and coefficients into a more concise form, making it easier to understand and use.

In the case of \((x+1)^4\), simplification involves combining like terms and applying basic arithmetic to achieve the simplest form. Here’s how it looks:

• \(x^4\) remains \(x^4\) because it's a distinct term
• \(4x^3\) is already simplified
• \(6x^2\) again has no other terms to combine with
• \(4x\) also stands alone
• \(1\) is just \(1\)

With no like terms to concatenate further, the expression is as straightforward as it gets. Therefore, the equivalent expression becomes \(x^4 + 4x^3 + 6x^2 + 4x + 1\). The process not only reveals individual term weights but also conclusively offers a clear polynomial representation of the original binomial expression.

Simplifying is crucial for solving equations, graphing, and further mathematical manipulation of the expression. It provides a cleaner, more usable form that eases further interpretation.