Question

Expand each expression using the Binomial Theorem. $$ (x+3)^{5} $$

Short answer

The expansion is \( x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243 \).

Step-by-step solution

Identify the Binomial

Recognize that the binomial to be expanded is \( (x+3)^5 \).

Write the Binomial Theorem Formula

Recall that the Binomial Theorem is given by \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \].

Apply the Values into the Binomial Theorem

Substitute \( a = x \), \( b = 3 \), and \( n = 5 \) into the formula. It becomes \[ (x+3)^5 = \sum_{k=0}^{5} \binom{5}{k} x^{5-k} 3^k \].

Calculate Each Term Separately

Expand and calculate each term using the binomial coefficients: - When \( k = 0 \): \[ \binom{5}{0} x^5 3^0 = 1 \times x^5 \times 1 = x^5 \]- When \( k = 1 \): \[ \binom{5}{1} x^4 3^1 = 5 \times x^4 \times 3 = 15x^4 \]- When \( k = 2 \): \[ \binom{5}{2} x^3 3^2 = 10 \times x^3 \times 9 = 90x^3 \]- When \( k = 3 \): \[ \binom{5}{3} x^2 3^3 = 10 \times x^2 \times 27 = 270x^2 \]- When \( k = 4 \): \[ \binom{5}{4} x^1 3^4 = 5 \times x \times 81 = 405x \]- When \( k = 5 \): \[ \binom{5}{5} x^0 3^5 = 1 \times 1 \times 243 = 243 \]

Combine All Terms

Sum all the terms obtained: \[ x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243 \].

Key concepts

Expanding Binomials

The process of expanding binomials allows us to transform an expression like \( (x+3)^5 \) into a polynomial. The key tool for this transformation is the Binomial Theorem. By applying this theorem, any binomial expression raised to a power can be expanded systematically.

When expanding binomials, the original binomial expression must be identified. In our case, we start with \( (x+3)^5 \). This is our binomial to expand. Using the Binomial Theorem simplifies the expansion as it provides a predetermined structure for each term in the polynomial.

The theorem outlines that each term will be the product of a binomial coefficient, a power of the first term of the binomial (here, \( x \)), and a power of the second term (here, \( 3 \)).

Binomial Coefficients

Binomial coefficients are essential in the application of the Binomial Theorem. Each coefficient quantifies how many ways a term in the expansion can be achieved. They are represented by \( \binom{n}{k} \), where \( n \) is the total power to which the binomial is raised, and \( k \) refers to the specific term in the sequence being expanded.

These coefficients can be obtained using the combination formula \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]. In our specific problem of expanding \( (x+3)^5 \), we calculate the binomial coefficients for \( n = 5 \) and \( k \) values from 0 to 5:

• For \( k = 0, \binom{5}{0} = 1 \).
• For \( k = 1, \binom{5}{1} = 5 \).
• For \( k = 2, \binom{5}{2} = 10 \).
• For \( k = 3, \binom{5}{3} = 10 \).
• For \( k = 4, \binom{5}{4} = 5 \).
• For \( k = 5, \binom{5}{5} = 1 \).

These coefficients will multiply with the respective powers of \( x \) and \( 3 \) to give each term's value in the expanded form.

Polynomial Expansion

Polynomial expansion involves turning a binomial raised to a power into a sum of individual terms. Each term is a combination of coefficients and variables raised to different powers. Using the Binomial Theorem can simplify this potentially complex process.

For our example, \( (x+3)^5 \), the polynomial expanded form is derived by calculating each term separately as shown:

• Term 1: \( \binom{5}{0} x^5 3^0 = x^5 \).
• Term 2: \( \binom{5}{1} x^4 3^1 = 15x^4 \).
• Term 3: \( \binom{5}{2} x^3 3^2 = 90x^3 \).
• Term 4: \( \binom{5}{3} x^2 3^3 = 270x^2 \).
• Term 5: \( \binom{5}{4} x^1 3^4 = 405x \).
• Term 6: \( \binom{5}{5} x^0 3^5 = 243 \).

Adding these terms together yields the final polynomial: \( x^5 + 15x^4 + 90x^3 + 270x^2 + 405x + 243 \).

The process of polynomial expansion can thus transform a compact binomial expression into detailed and spread out polynomial form.

Combinatorics

Combinatorics plays a crucial role in understanding binomial expansion. It helps in calculating the binomial coefficients which count the number of ways to arrange or combine items.

Each term in the polynomial expansion of a binomial is influenced by combinations, as illustrated by \[ \binom{n}{k} \], where \( n \) denotes the total number of items (or tasks), and \( k \) denotes the number of selections.

Combinations reflect the binomial theorem's logic as follows:
\[ \binom{5}{0} = 1, \binom{5}{1} = 5, \binom{5}{2} = 10, \binom{5}{3} = 10, \binom{5}{4} = 5, \binom{5}{5} = 1 \].

These values show the different ways elements can combine in any given order to form the expanded polynomial. Understanding combinatorics empowers you to easily compute and apply these combinations in binomial expansions.