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Magazine Chapter 12: Problem 25
Find the first three terms in the expansion of \((x+2 y)^{20}\)
Short Answer
Expert verified
The first three terms are \(x^{20} + 40x^{19}y + 760x^{18}y^2\).
Step by step solution
01
Identify the Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form \((a+b)^n\). It is given by:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\]where \(\binom{n}{k}\) is the binomial coefficient.
02
Determine Values for Binomial Expansion
For our expression \((x+2y)^{20}\), identify \(a = x\), \(b = 2y\), and \(n = 20\). We'll use these in the binomial expansion formula.
03
Calculate the First Term
The first term occurs when \(k = 0\). Substitute into the formula:\[\binom{20}{0}x^{20-0}(2y)^{0} = x^{20}\]Thus, the first term is \(x^{20}\).
04
Calculate the Second Term
The second term occurs when \(k = 1\). Substitute into the formula:\[\binom{20}{1}x^{20-1}(2y)^{1} = 20x^{19}(2y) = 40x^{19}y\]Thus, the second term is \(40x^{19}y\).
05
Calculate the Third Term
The third term occurs when \(k = 2\). Substitute into the formula:\[\binom{20}{2}x^{20-2}(2y)^{2} = \frac{20 \times 19}{2}x^{18}(4y^2)\]\[= 190x^{18} imes 4y^2 = 760x^{18}y^2\]Thus, the third term is \(760x^{18}y^2\).
06
Compile the Result
The first three terms in the expansion of \((x+2y)^{20}\) are \(x^{20} + 40x^{19}y + 760x^{18}y^2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Expansion
The Binomial Expansion is a method used to expand expressions that contain a sum raised to a power. This kind of expression is known as a binomial expression. For example, \((x + 2y)^{20}\) is a binomial expression where two terms, \(x\) and \(2y\), are raised to the power of 20.
To expand this expression, we utilize the Binomial Theorem. This theorem provides a convenient formula to expand any binomial expression of the form \((a + b)^n\).
Here's how the Binomial Theorem works in a nutshell:
To expand this expression, we utilize the Binomial Theorem. This theorem provides a convenient formula to expand any binomial expression of the form \((a + b)^n\).
Here's how the Binomial Theorem works in a nutshell:
- Identify \(a\), \(b\), and \(n\) in the expression.
- Use the formula given by the theorem: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\).
- Calculate each term separately by plugging the values into the formula.
Binomial Coefficients
Binomial Coefficients are a central component of the Binomial Theorem. These coefficients are represented in the form \(\binom{n}{k}\), also known as 'n choose k'. They play a significant role in determining the weight of each term in a binomial expansion.
The formula for calculating these coefficients is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where '!' denotes a factorial.
This formula calculates the number of ways to choose \(k\) elements from \(n\) elements, which is why they are called coefficients.
The formula for calculating these coefficients is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where '!' denotes a factorial.
This formula calculates the number of ways to choose \(k\) elements from \(n\) elements, which is why they are called coefficients.
- When \(k = 0\), \(\binom{n}{0} = 1\), because there is one way to choose no elements.
- Example: For \(n = 20\) and \(k = 2\), the binomial coefficient \(\binom{20}{2}\) is \[\frac{20 \times 19}{2 \times 1} = 190\] This will factor into the term of the binomial expansion.
- These coefficients increase and decrease in a symmetric pattern known as Pascal's Triangle.
Polynomial Expansion
Polynomial Expansion involves expressing a polynomial expression as a sum of individual terms. In the context of binomials, it means breaking down expressions like \((x + 2y)^{20}\) into a more manageable sum of terms.
This expansion is expressed as a series, each term of which is determined by the:
This expansion is expressed as a series, each term of which is determined by the:
- Binomial coefficient for that specific term position.
- Powers of the terms in the binomial, which decrease for one part and increase for another as you progress from term to term.
- The first term is purely \(x\), to the power of 20: \(x^{20}\).
- The second term incorporates the next level of complexity: \(40x^{19}y\).
- The third term adds more with \(760x^{18}y^2\).
