Problem 30 - Algebra and Trigonometry

Question

Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{3}\) in the expansion of \((x-3)^{10}\)

Step-by-Step Solution

Answer

-262440

Step 1 - Write the general form of the Binomial Theorem

The Binomial Theorem states that the expansion of \((a + b)^n\) is given by:\[ (a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \ldots + \binom{n}{n}a^0 b^n \] where \( \binom{n}{k} \) is the binomial coefficient.

Step 2 - Identify the values of a, b, and n

For the expression \((x-3)^{10}\), we identify \(a = x\), \(b = -3\), and \(n = 10\).

Step 3 - Determine the term containing \(x^3\)

In the binomial expansion, the general term is given by \( \binom{n}{k} a^{n-k} b^k \). To find the term containing \(x^3\), set the exponent of \(x\) in \(a^{n-k} = x^{n-k}\) equal to 3. Thus, \(n-k = 3\).

Step 4 - Solve for k

Given \(n - k = 3\), we substitute \(n = 10\): \10 - k = 3 \Rightarrow k = 7\.

Step 5 - Substitute and calculate the coefficient

Using \(k = 7\), the term is \( \binom{10}{7} x^{10-7} (-3)^7 \). Simplify this to get \( \binom{10}{7} x^3 (-3)^7 \).

Step 6 - Compute the binomial coefficient and power

Calculate \( \binom{10}{7} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = 120 \). Calculate \( (-3)^7 = -2187 \).

Step 7 - Multiply to find the coefficient

The coefficient of \(x^3\) is thus \( 120 \times (-2187) = -262440 \).

Key Concepts

Binomial Expansion

When dealing with polynomial expressions raised to a power, the Binomial Theorem is a powerful tool.
It allows us to expand expressions of the form \((a + b)^n\) into a sum of terms involving binomial coefficients.
The general formula for the binomial expansion is given by:\[(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \ldots + \binom{n}{n}a^0 b^n\]Here's a breakdown of the notation and what everything means:

• \(a\) and \(b\) are any numbers or variables.
• \(n\) is the power to which the binomial is raised.
• \(\binom{n}{k}\) represents the binomial coefficient, which we’ll explain in detail below.

By applying this theorem, we can find a specific term in the expansion without having to fully expand the binomial. This is particularly useful for large powers.

Binomial Coefficient

Binomial coefficients are a key component in the binomial expansion formula.
They are denoted as \(\binom{n}{k}\) and read as 'n choose k'.
A binomial coefficient tells us how many ways we can choose k elements from a set of n elements without regard to the order.
Mathematically, it's calculated using factorials as follows:\[\binom{n}{k} = \frac{n!}{k! (n - k)!}\]The exclamation mark denotes factorial, which means the product of all positive integers up to that number.
For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

• \(\binom{10}{7} = \binom{10}{3} = \frac{10!}{3!(10 - 3)!} = 120\) in our example, since choosing 7 out of 10 is essentially the same as choosing 3 out of 10.

Understanding binomial coefficients enables us to determine the coefficients of individual terms in any binomial expansion.

Coefficients in Polynomial

In any polynomial expression, coefficients are the numerical factors that accompany the variable terms.
For instance, in the term \(5x^3\), 5 is the coefficient.
When using the binomial theorem to expand \((x-3)^{10}\), it helps to break down the steps:

• Identify \(a\), \(b\), and \(n\) where, in this case, \(a = x\), \(b = -3\), and \(n = 10\).
• Use the general term \(\binom{n}{k} a^{n-k} b^k\) to find the desired term. Setting the exponent of \(x\) to 3 gives us \(n - k = 3\) or \(k = 7\).
• Calculate the binomial coefficient \(\binom{10}{7}\) which simplifies to 120.
• Raise \(b\) to the power of \(k\), so \((-3)^7 = -2187\).
• Multiply the binomial coefficient by this power, resulting in the term's coefficient: \(120 \times -2187 = -262440\).

Through these steps, we derive the coefficient for any specified term in the binomial expansion.