Problem 32 - College Algebra

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+3)^{8} $$

Step-by-Step Solution

Answer

\[6,561 + 17,496x + 20,412x^2\]

Step 1: Identify the variables in the binomial

The binomial in question is \((x+3)^8\). Here, 'x' is 'a', '3' is 'b' and '8' is 'n' in the general form of the binomial expression \((a+b)^n\).

Step 2: Calculate the coefficients using Binomial Theorem

The binomial theorem says the expansion of \((a+b)^n\) is characterized by the formula \[nCk * a^k * b^{n-k}\], where 'k' is the term number minus 1 (since we start counting from 0). So, for the first three terms where k = 0, 1, 2, we should plug in these values and calculate the formula. For k=0, it's \[8C0 * x^0 * 3^8\]. For k=1, it's \[8C1 * x^1 * 3^7\]. And for k=2, it's \[8C2 * x^2 * 3^6\].

Step 3: Simplify the coefficients

Simplify each of these expressions using the known formula for binomial coefficients. So the three terms are: \[8C0 * x^0 * 3^8 = 1 * 1 * 6,561 = 6,561\], \[8C1 * x^1 * 3^7 = 8 * x * 2,187 = 17,496x\], and \[8C2 * x^2 * 3^6 = 28 * x^2 * 729 = 20,412x^2\]. Put these together to get the first three terms in the binomial expansion.

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental mathematical principle that explains how to expand expressions of the form \((a+b)^n\). It provides a powerful tool for algebraic expression where powers are involved. The theorem states that each term in the expansion can be determined using the formula:

• \[nCk * a^{n-k} * b^k\]

Here, \(nCk\) stands for the binomial coefficient, which represents the number of ways to choose \(k\) items from \(n\) items without considering the order.
For example, in the expression \((x+3)^8\), the binomial theorem helps us expand this to more manageable smaller terms. Each term in this expansion is computed using a systematic application of the theorem, simplifying the expression significantly.

Binomial Coefficients

Binomial coefficients play a crucial part in the binomial theorem. These coefficients are the numerical multipliers of the terms in a binomial expansion. You can find binomial coefficients using the formula:

• \[nCk = \frac{n!}{k!(n-k)!}\]

Where \(!\) denotes a factorial, which is the product of an integer and all the integers below it. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
In our exercise, evaluating the binomial coefficient for each term involves getting these combinations:

• \(8C0 = 1\)
• \(8C1 = 8\)
• \(8C2 = 28\)

These coefficients help multiply the corresponding powers of \(a\) and \(b\), making the expansion straightforward.

Algebraic Expressions

Understanding algebraic expressions is vital when dealing with binomial expansions. An algebraic expression is a combination of variables, numbers, and operations such as addition or multiplication.

• Variables (e.g., \(x\), \(y\)) can represent numbers and are often denoted by letters.
• Numbers are the constants in expressions (e.g., \(3\), \(8\)).
• Operations refer to the arithmetic actions between numbers and variables, such as addition, subtraction, multiplication, or division.

In \((x+3)^8\), \(x\) and \(3\) are part of a simple yet powerful algebraic expression. The value \(x\) can change depending on the problem, while \(3\) remains constant. This flexibility and combination of stable and variable elements enable complex calculations possible. After applying the binomial theorem, the expression simplifies into separate terms, making it easier to interpret results or solve problems that, initially, appeared complex.