Problem 25 - Algebra and Trigonometry

Question

Find the first three terms in the expansion of \((x+2 y)^{20}\)

Step-by-Step Solution

Answer

The first three terms are \(x^{20} + 40x^{19}y + 760x^{18}y^2\).

Step 1: 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.

Step 2: 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.

Step 3: 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 \(x^{20}\).

Step 4: 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 \(40x^{19}y\).

Step 5: 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 \(760x^{18}y^2\).

Step 6: Compile the Result

The first three terms in the expansion of \((x+2y)^{20}\) are \(x^{20} + 40x^{19}y + 760x^{18}y^2\).

Key Concepts

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:

• 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.

Using this method makes solving binomial expressions systematic and straightforward. Students can quickly find any number of terms from the expansion.

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.

• 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.

Understanding binomial coefficients is crucial, as they directly influence each term's magnitude in a binomial expansion.

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:

• 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.

For example, in the expansion of \((x + 2y)^{20}\):

• 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 \(760x^{18}y^2\).

Polynomial expansion simplifies the expression by breaking it into these distinct, individual terms. Understanding how to follow through the polynomial expansion process simplifies complex calculations and helps in solving higher-level algebra and calculus problems.