Problem 26 - College Algebra

Question

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial. $$ (x-2 y)^{8} $$

Step-by-Step Solution

Answer

The first three terms are \(x^8 - 16x^7y + 112x^6y^2\).

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions of the form \((a + b)^n\). It states that: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Here, \(\binom{n}{k}\) is the binomial coefficient, \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). In our case, \(n = 8\), \(a = x\), and \(b = -2y\).

Step 2: Calculate the First Term

To find the first term when \(k=0\): \[\begin{align*}\text{First Term} &= \binom{8}{0} x^{8} (-2y)^{0} \&= 1 \cdot x^{8} \cdot 1 \&= x^{8}\end{align*}\]

Step 3: Calculate the Second Term

To find the second term when \(k=1\): \[\begin{align*}\text{Second Term} &= \binom{8}{1} x^{7} (-2y)^{1} \&= 8 \cdot x^{7} \cdot (-2y) \&= -16x^{7}y\end{align*}\]

Step 4: Calculate the Third Term

To find the third term when \(k=2\): \[\begin{align*}\text{Third Term} &= \binom{8}{2} x^{6} (-2y)^{2} \&= 28 \cdot x^{6} \cdot 4y^2 \&= 112x^{6}y^{2}\end{align*}\]

Step 5: Combine the First Three Terms

We can now combine the results from steps 2-4 to find the first three terms of the binomial expansion:\[x^{8} - 16x^{7}y + 112x^{6}y^{2}\]

Key Concepts

Binomial Expansion

The binomial expansion is a crucial concept in algebra, particularly when dealing with expressions raised to a power. It allows a binomial, which is an expression with two terms (for example,
\((a + b)^n\)), to be expanded into a polynomial consisting of several terms.
The Binomial Theorem provides the formula for this expansion, stating that:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
This displays the transformation of an expression into a series of terms, with each term involving a combination of both variables, \(a\) and \(b\), raised to various powers. The exponents of \(a\) and \(b\) in each term always sum to \(n\).
This process is useful, for instance, when calculating probabilities, solving combinatorial problems, or simply simplifying algebraic expressions for easier handling or integration.

Binomial Coefficient

In the binomial expansion, binomial coefficients play an essential role as they determine the weights of each term. The binomial coefficient, denoted as \(\binom{n}{k}\), indicates the number of ways to choose \(k\) elements from a total of \(n\) elements, and it can be calculated using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
Here, the factorial notation \(!\) describes the product of all positive integers up to that number. In a binomial expansion, each term of \((a + b)^n\) is multiplied by a binomial coefficient.
For example, during the expansion of \((x - 2y)^8\), the coefficients for the terms when \(k=0, 1, 2\) can be calculated as \(\binom{8}{0}, \binom{8}{1}, \binom{8}{2}\).
These correspond to the numerical components 1, 8, and 28, used to find the first three terms of the expansion. Understanding these coefficients is key to mastering the binomial theorem and its applications.

Polynomial Terms

After applying the Binomial Theorem, the result is a polynomial, which is a sum of multiple terms. Each term in this polynomial is a product involving powers of the variables from the original binomial expression.
For instance, in the expansion of \((x - 2y)^8\), the polynomial terms formed are \(x^8\), \(-16x^7y\), and \(112x^6y^2\).
These terms can be understood as:

• \(x^8\): the simplest term, derived entirely from \(x\).
• \(-16x^7y\): combines powers of \(x\) and \(y\), reflecting the effects of their multiplication.
• \(112x^6y^2\): a more complex term that includes higher powers of both \(x\) and \(y\).

Such terms collectively represent the original raised binomial, spread out in an expanded form. Grasping the structure of these polynomial terms helps in solving and understanding algebraic equations efficiently.