Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

In the expression of \((x+y)^{3},\) what is the sum of the powers of the third term?

Step-by-Step Solution

Answer

The sum of the powers of x and y in the third term of the expansion \((x+y)^3\) is 3.

Step 1: Understand the Problem

The binomial \((x+y)^{3}\) expands into four terms. We need to find out the powers of x and y in the third term.

Step 2: Apply Binomial Theorem

According to the binomial theorem, each term in the expansion of a binomial expression \((x+y)^n\) can be expressed in the form \({n\choose k} x^{n-k}y^k\), where k is the term number (starts from 0). Here n is 3, so in the third term the value of k will be 2.

Step 3: Find the Powers of X and Y

For the third term (k=2), the powers of x and y would be found using formulas \(n - k\) and \(k\) respectively. So the power of x will be \(3 - 2 = 1\) and the power of y will be \(2\).

Step 4: Sum of the Powers

The sum of the powers of x and y in the third term is therefore \(1 + 2 = 3\).

Key Concepts

Binomial Expansion

The binomial expansion is a fundamental concept in algebra. It allows us to expand expressions of the form \((x + y)^n\) into a series of terms. This expansion is crucial because it breaks down complex expressions into simpler components.
Using the binomial theorem, each term in the expansion can be written as \({n\choose k} x^{n-k}y^k\), where \(n\) is the exponent in the original binomial, and \(k\) represents the term number starting from zero.
Here are some key features of the binomial expansion:

• The number of terms in the expansion is \(n + 1\).
• For example, \((x+y)^3\) expands into four terms.
• Each term has a coefficient that is a binomial coefficient \({n\choose k}\).

By understanding how to expand a binomial, we can predict and calculate values within these expressions.

Powers of Variables

In any algebraic expression, the powers of variables like \(x\) and \(y\) are critical. The powers tell us how many times a variable is multiplied by itself. During the binomial expansion process, each term has specific powers for \(x\) and \(y\).
For the binomial expression \((x+y)^n\), every term will have the power of \(x\) as \(n-k\) and the power of \(y\) as \(k\), where \(k\) changes with each term of the expansion. In our example \((x+y)^3\), the third term's (using \(k=2\)) power of \(x\) is calculated as \(3-2=1\), and the power of \(y\) is \(2\).
Understanding the powers of variables helps in determining the contribution of each variable within any term of the expansion.

Sum of Powers

The sum of the powers in a term within a binomial expansion provides valuable insights into the structure of expressions. It is simply the addition of the powers of \(x\) and \(y\) for each term.
In the expansion of \((x+y)^3\), for the third term where \(k=2\), the power of \(x\) is \(1\) and the power of \(y\) is \(2\). Therefore, the sum of the powers is \(1+2=3\).
Knowing the sum of the powers can help verify the consistency of the terms within the overall expression. Thus, this is a useful check that you have correctly identified the powers within the term.