Problem 11 - Algebra for College Students

Question

Use the binomial theorem to expand each binomial. $$(x+1)^{3}$$

Step-by-Step Solution

Answer

The expansion is \( x^3 + 3x^2 + 3x + 1 \)

Step 1: Write the Binomial Theorem

The binomial theorem states that: \[ (a + b)^n = \ \ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. Here, we have \( (x+1)^3 \), so identify \( a = x \), \( b = 1 \), and \( n = 3 \)

Step 2: Identify the Terms

For each term in the expansion, use the binomial coefficient formula: \[ \binom{n}{k} = \frac{n!}{k! (n-k)!} \]. Substitute \( n = 3 \) and calculate for \( k = 0, 1, 2, 3 \)

Step 3: Calculate Each Factor

Calculate the following using the binomial coefficient: \[ \binom{3}{0} = 1 \], \[ \binom{3}{1} = 3 \], \[ \binom{3}{2} = 3 \], \[ \binom{3}{3} = 1 \]

Step 4: Write Each Term

Write the terms using the coefficients and the powers of \( a = x \) and \( b = 1 \): \[ \binom{3}{0} x^{3-0} 1^{0} = x^3 \] \[ \binom{3}{1} x^{3-1} 1^{1} = 3x^2 \] \[ \binom{3}{2} x^{3-2} 1^{2} = 3x \] \[ \binom{3}{3} x^{3-3} 1^{3} = 1 \]

Step 5: Combine the Terms

Add all the terms together to get the expansion: \[ x^3 + 3x^2 + 3x + 1 \]

Key Concepts

Binomial Expansion

The binomial expansion involves breaking down a binomial expression raised to a power into a sum of several terms. The binomial theorem is key here. It states: \((a + b)^n\) can be expanded using: The binomial expansion is handy in algebra and calculus. It turns complicated expressions into simpler, manageable sums.

Binomial Coefficients

Binomial coefficients are vital in the binomial theorem. They indicate the number of ways to choose elements from a set. The formula to find a binomial coefficient is: Beta functions are the basis of binomial coefficients. For the example \((x+1)^3\), we need the coefficients:

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

Understanding these makes expanding binomials much easier.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial as the sum of its monomial (single term) components. When using the binomial theorem, the polynomial: Combine these terms to get: Polynomial expansion helps in simplifying expressions and solving algebraic problems. It's a fundamental tool in algebra.