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Magazine Chapter 6: Problem 5
$$ \text { Expand the expressions in 1-9 using the binomial theorem. } $$ $$ (p-2 q)^{4} $$
Short Answer
Expert verified
The expanded expression using the binomial theorem is: \((u^2 - 3v)^4 = u^8 - 12u^6v + 54u^4v^2 - 108u^2v^3 + 81v^4\).
Step by step solution
01
Identify the variables in the expression
In the given expression \((u^2 - 3v)^4\), we have \(a = u^2\), \(b = -3v\), and \(n = 4\).
02
Apply the binomial theorem
Using the binomial theorem, we can expand the expression as:
\[(u^2 - 3v)^4 = \sum_{k=0}^4 \binom{4}{k} (u^2)^{4-k} (-3v)^k\]
Now, we will calculate each term in the expansion.
03
Calculate the binomial coefficients
We need to find the binomial coefficients for each term in the expansion:
\[\binom{4}{0} = \frac{4!}{0!4!} = 1\]
\[\binom{4}{1} = \frac{4!}{1!3!} = 4\]
\[\binom{4}{2} = \frac{4!}{2!2!} = 6\]
\[\binom{4}{3} = \frac{4!}{3!1!} = 4\]
\[\binom{4}{4} = \frac{4!}{4!0!} = 1\]
04
Calculate the powers of the variables in each term
Now, we will compute the powers of \(u^2\) and \((-3v)\) in each term:
\(k=0\): \((u^2)^{4-0} (-3v)^0 = u^8\)
\(k=1\): \((u^2)^{4-1} (-3v)^1 = u^6 (-3v)\)
\(k=2\): \((u^2)^{4-2} (-3v)^2 = u^4 (9v^2)\)
\(k=3\): \((u^2)^{4-3} (-3v)^3 = u^2 (-27v^3)\)
\(k=4\): \((u^2)^{4-4} (-3v)^4 = (81v^4)\)
05
Combine the binomial coefficients with the powers of the variables
Now, we will multiply the binomial coefficients by the powers of the variables for each term:
\(k=0\): \(\binom{4}{0} u^8 = 1 \cdot u^8 = u^8\)
\(k=1\): \(\binom{4}{1} u^6 (-3v) = 4 \cdot u^6 (-3v) = -12u^6v\)
\(k=2\): \(\binom{4}{2} u^4 (9v^2) = 6 \cdot u^4 (9v^2) = 54u^4v^2\)
\(k=3\): \(\binom{4}{3} u^2 (-27v^3) = 4 \cdot u^2 (-27v^3) = -108u^2v^3\)
\(k=4\): \(\binom{4}{4} (81v^4) = 1 \cdot (81v^4) = 81v^4\)
06
Write the final expanded expression
Finally, we will put all the terms together to obtain the expanded expression:
\((u^2 - 3v)^4 = u^8 - 12u^6v + 54u^4v^2 - 108u^2v^3 + 81v^4\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Expansion
When you see an expression like \(u^2 - 3v)^4\), you are looking at a polynomial raised to a power. Expanding this using the binomial theorem reveals the various terms that result when you multiply this polynomial by itself four times. In our example, polynomial expansion simplifies a complex multiplication problem into a series of easier calculations, breaking the process into individual terms involving powers of \(u^2\) and \(3v\), and combining them to give the final expanded form.
Understanding polynomial expansion is crucial for many areas of mathematics, including algebra, calculus, and beyond. It allows us to manipulate expressions more easily and solve equations that would otherwise be too laborious to handle manually. The ability to expand polynomials is not just an academic exercise but a powerful tool in the problem-solving arsenal for anyone studying advanced levels of mathematics.
Understanding polynomial expansion is crucial for many areas of mathematics, including algebra, calculus, and beyond. It allows us to manipulate expressions more easily and solve equations that would otherwise be too laborious to handle manually. The ability to expand polynomials is not just an academic exercise but a powerful tool in the problem-solving arsenal for anyone studying advanced levels of mathematics.
Binomial Coefficients
Binomial coefficients are the numbers that appear in the binomial expansion. They are denoted by \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from a set of \(n\) elements. In the context of our binomial theorem journey, these numbers reveal how each term in the expansion is weighted.
For example, when expanding \(u^2 - 3v)^4\), the coefficients \(\binom{4}{0}\), \(\binom{4}{1}\), up to \(\binom{4}{4}\) determine the contribution of each term of the polynomial after the expansion. These coefficients form a symmetrical pattern, often referred to as Pascal's Triangle, and they are crucial for understanding how the variables \(u^2\) and \(3v\) combine and contribute to each term of the expanded polynomial.
For example, when expanding \(u^2 - 3v)^4\), the coefficients \(\binom{4}{0}\), \(\binom{4}{1}\), up to \(\binom{4}{4}\) determine the contribution of each term of the polynomial after the expansion. These coefficients form a symmetrical pattern, often referred to as Pascal's Triangle, and they are crucial for understanding how the variables \(u^2\) and \(3v\) combine and contribute to each term of the expanded polynomial.
Factorial Notation
Factorial notation is used in mathematics as a convenient way to express the product of an integer and all the positive integers below it. It's represented by an exclamation mark \(n!\), where \(n\) is any non-negative integer. For instance, \((3!)\) would be calculated as \((3 \times 2 \times 1\)).
Factorials are particularly important in combinatorics, probability, and, as highlighted in our discussion, the computation of binomial coefficients. In the context of the binomial theorem, factorials allow us to easily calculate the weights \(\binom{n}{k}\), which are essential for determining the proportions of each term of our polynomial expansion. By understanding the role of factorial notation, anyone exploring the binomial theorem can adeptly navigate through the computations required for polynomial expansions.
Factorials are particularly important in combinatorics, probability, and, as highlighted in our discussion, the computation of binomial coefficients. In the context of the binomial theorem, factorials allow us to easily calculate the weights \(\binom{n}{k}\), which are essential for determining the proportions of each term of our polynomial expansion. By understanding the role of factorial notation, anyone exploring the binomial theorem can adeptly navigate through the computations required for polynomial expansions.
