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Magazine Chapter 12: Problem 25
Use the Binomial Theorem to expand the expression. $$(x+2 y)^{4}$$
Short Answer
Expert verified
The expanded form is \(x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4\).
Step by step solution
01
Identify Binomial Components
The expression \((x + 2y)^4\) is a binomial raised to a power. Here, the binomial components are \(a = x\) and \(b = 2y\), and the exponent \(n = 4\).
02
Recall the Binomial Theorem Formula
The Binomial Theorem states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For this problem, substitute \(n = 4\), \(a = x\), and \(b = 2y\) into the formula.
03
Calculate Each Binomial Coefficient
Calculate the binomial coefficients \(\binom{4}{k}\) for each \(k\) from 0 to 4. These are: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).
04
Expand Using the Binomial Coefficients
Substitute the binomial coefficients into the expansion: \((x+2y)^4 = \binom{4}{0}x^4(2y)^0 + \binom{4}{1}x^3(2y)^1 + \binom{4}{2}x^2(2y)^2 + \binom{4}{3}x^1(2y)^3 + \binom{4}{4}x^0(2y)^4\).
05
Simplify Each Term
Evaluate and simplify each term from the expansion:- \(1 \, \cdot \, x^4 \, \cdot \, 1 = x^4\)- \(4 \, \cdot \, x^3 \, \cdot \, 2y = 8x^3y\)- \(6 \, \cdot \, x^2 \, \cdot \, (2y)^2 = 24x^2y^2\)- \(4 \, \cdot \, x \, \cdot \, (2y)^3 = 32xy^3\)- \(1 \, \cdot \, 1 \, \cdot \, (2y)^4 = 16y^4\).
06
Write the Full Expansion
Combine the simplified terms to get the expanded form: \(x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Coefficients
In mathematics, binomial coefficients are a fundamental concept in algebra, specifically when working with binomials and their expansions. These coefficients are represented as \( \binom{n}{k} \), known as "n choose k." They are integral in determining each term of a polynomial expansion of a binomial.
- The formula for a binomial coefficient is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, \( k \) is the number of items being chosen, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.
- For example, in the expansion of \( (x + 2y)^4 \), we calculate the coefficients \( \binom{4}{0}, \binom{4}{1}, \ldots, \binom{4}{4} \), which are respectively 1, 4, 6, 4, and 1.
- These numbers help us determine the weights of the terms in the binomial expansion.
Polynomial Expansion
Polynomial expansion is the process of expressing a binomial expression raised to a power as a sum of terms. It involves breaking down the expression into a series of simpler polynomials. The Binomial Theorem offers a straightforward method for achieving this.
- Using the theorem, we can expand \((a + b)^n\) into a sum involving binomial coefficients, powers of \(a\), and powers of \(b\). This expansion is represented as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).
- For instance, when we apply it to \((x + 2y)^4\), we expand it into a polynomial consisting of terms like \(x^4, 8x^3y,\) and others.
- Each term in this expanded form is derived from multiplying a binomial coefficient with appropriate powers of \(a\) and \(b\).
Exponentiation
Exponentiation is a mathematical operation involving numbers or expressions being raised to a power. This operation is central to binomial expansion, where the binomial expression is raised to an exponent, often a positive integer.
- In the context of the Binomial Theorem, exponentiation determines the power to which the binomial components \(a\) and \(b\) are raised in each term of the expansion.
- In the example \((x + 2y)^4\), the term \(x^4\) results from raising \(x\) to the power of four, illustrating exponentiation in its simplest form.
- Exponentiation impacts both numbers and variables. For instance, \((2y)^3\) involves raising 2 and \(y\) to the same power simultaneously, resulting in \(8y^3\).
