Problem 20 - Algebra for College Students

Question

Use the binomial theorem to expand each binomial. $$(a+3 b)^{4}$$

Step-by-Step Solution

Answer

Use the binomial theorem In this particular example we have a+b+c

Step 1 - Identify Components of the Binomial Theorem

The general form of the binomial theorem is given by: decomposing the given binomial term into the general formon the power terms equating the known terms of the binomial theorem to examining the expected poweridentifying the greater than expected results and further code in LaTeX ...$$ $$ e^{i \theta} = \frac{{1}}{{n}} P^{i}_{i \theta} (2)$$

Step 2 - Write the Binomial Theorem

The binomial theorem states that ...}, {

Key Concepts

Binomial Expansion

The binomial expansion allows us to expand expressions of the form \( (a + b)^n \) using the binomial theorem. This theorem provides a formula to expand binomials into a sum of terms involving coefficients, powers of the first term, and powers of the second term. For the binomial expression \( (a + 3b)^4 \), we use the binomial theorem to find each term. It helps simplify and calculate powers of binomials more easily. The general sum for the expansion is given by: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \]. Here, \( a \), \( b \), and \( n \) are the components of the binomial.

Exponents

Exponents are used to describe how many times a number (the base) is multiplied by itself. In \( (a + 3b)^4 \), 4 is the exponent, indicating that \( a + 3b \) should be multiplied by itself four times. The binomial theorem recognizes patterns in these multiplications and utilizes them for simplification. When expanding the binomial, exponents of each term decline for the first term \( a \) and increase for the second term \( 3b \) as we move from term to term.

Binomial Coefficients

The coefficients in the binomial expansion are known as binomial coefficients. They are represented as \( \binom{n}{k} \). For example, \( \binom{4}{2} \) can be read as '4 choose 2'. These coefficients represent the number of ways to choose \( k \) elements from \( n \) elements and are found using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \], where \( n! \) (n factorial) means multiplying all positive integers up to \( n \). So, in our example \( (a + 3b)^4 \), coefficients are 1, 4, 6, 4, and 1.

Combinatorics

Combinatorics is the field of mathematics concerning the counting, arrangement, and combination of objects. Binomial coefficients are deeply connected to combinatorial principles. When performing a binomial expansion, we are essentially dealing with combinations of different powers of the terms. In the expression \( (a + 3b)^4 \), each term of the expansion arises from different combinations of \( a \) and \( 3b \), as per the rules of combinatorics. These principles simplify the process, ensuring that we cover all possible combinations efficiently.