Question

In Exercises 51-58, use the Binomial Theorem to write the binomial expansion. \((x+2)^3\)

Short answer

The binomial expansion of \((x+2)^3\) is \(x^3+6x^2+12x+8\).

Step-by-step solution

Understanding the Binomial Theorem

The Binomial theorem is given by: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\] where n is a nonnegative integer called the index, a and b are any real numbers, and \(\binom{n}{k}\) is a binomial coefficient. Here, a will be x, b will be 2, and n will be 3.

Applying the Binomial Theorem to Our Problem

Apply the binomial theorem by substituting the appropriate values into the equation. The first term is \(\binom{3}{0}x^{3}2^{0}\), the second term is \(\binom{3}{1}x^{2}2^{1}\), the third term is \(\binom{3}{2}x^{1}2^{2}\) and the fourth term is \(\binom{3}{3}x^{0}2^{3}\). When calculated, this gives: \(x^{3}+6x^{2}+12x+8\).

Simplifying the Expression

After fully expanding and simplifying the expression, you should end up with: \(x^3+6x^2+12x+8\).

Key concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions that are raised to a power in a systematic way. It is expressed in the formula: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\]Here, \( (a+b)^n \) represents a binomial raised to the n-th power. This theorem is especially useful when we deal with larger powers, as it provides a structured method to expand the expression.

• Each term in the expansion is a product of three factors:
• A binomial coefficient \( \binom{n}{k} \), which is a number that shows how many ways there are to choose \( k \) elements from a set of \( n \) elements.
• The first base \( a \) raised to the power of \( n-k \).
• The second base \( b \) raised to the power of \( k \).

Understanding the binomial theorem simplifies and enhances the process of expanding expressions, making it feasible to work with polynomials of higher degrees without manually multiplying each term repeatedly.

Binomial Coefficients

Binomial coefficients are key to the binomial theorem and are represented as \( \binom{n}{k} \), where:

• \( n \) is the degree of the binomial.
• \( k \) is the specific term in the binomial expansion.

These coefficients tell us the number of ways to choose \( k \) items from \( n \) items, which relates directly to combinatorial mathematics. The formula for the 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, \( 3! = 3 \times 2 \times 1 = 6 \). This coefficient gives weight to each term in the expanded form of the binomial. Knowing how to calculate these values is crucial for constructing each term in the polynomial expansion.

Polynomial Expansion

Polynomial expansion refers to expressing a power of a binomial as a sum of terms. In the case of \((x+2)^3\), applying the binomial theorem gives us:\[x^3 + 6x^2 + 12x + 8\]Each term in this expansion originates from the combination of binomial coefficients and the relevant powers of the binomial's components, \( x \) and \( 2 \).

• The first term \( x^3 \) comes from \( \binom{3}{0}x^3 \cdot 2^0 \).
• The second term \( 6x^2 \) is derived from \( \binom{3}{1}x^2 \cdot 2^1 \).
• The third term \( 12x \) results from \( \binom{3}{2}x^1 \cdot 2^2 \).
• The final term \( 8 \) comes from \( \binom{3}{3}x^0 \cdot 2^3 \).

This methodical expansion process allows us to see how each term contributes to the overall polynomial. Understanding polynomial expansion through the lens of the binomial theorem aids greatly in simplifying complex algebraic expressions.