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Magazine Chapter 13: Problem 30
Use the Binomial Theorem to find the indicated coefficient or term. The coefficient of \(x^{3}\) in the expansion of \((x-3)^{10}\)
Short Answer
Expert verified
-262440
Step by step solution
01
- 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.
02
- Identify the values of a, b, and n
For the expression \((x-3)^{10}\), we identify \(a = x\), \(b = -3\), and \(n = 10\).
03
- 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, \(n-k = 3\).
04
- Solve for k
Given \(n - k = 3\), we substitute \(n = 10\): \10 - k = 3 \Rightarrow k = 7\.
05
- Substitute and calculate the coefficient
Using \(k = 7\), the term is \( \binom{10}{7} x^{10-7} (-3)^7 \). Simplify this to get \( \binom{10}{7} x^3 (-3)^7 \).
06
- Compute the binomial coefficient and power
Calculate \( \binom{10}{7} = \binom{10}{3} = \frac{10!}{3!(10-3)!} = 120 \). Calculate \( (-3)^7 = -2187 \).
07
- Multiply to find the coefficient
The coefficient of \(x^3\) is thus \( 120 \times (-2187) = -262440 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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鈥檒l explain in detail below.
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 脳 4 脳 3 脳 2 脳 1 = 120.
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 脳 4 脳 3 脳 2 脳 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.
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:
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: \(120 \times -2187 = -262440\).
