Problem 12 - Algebra for College Students

Question

Use the binomial theorem to expand each binomial. $$(y+1)^{4}$$

Step-by-Step Solution

Answer

Expands to \( y^4 + 4y^3 + 6y^2 + 4y + 1 \).

Step 1: State the Binomial Theorem

The binomial theorem states that for any positive integer n, \((a + b)^n\) can be expanded as: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \].

Step 2: Identify Values

In the expression \((y + 1)^4\), identify the values: \(a = y\), \(b = 1\), and \(n = 4\).

Step 3: Apply the Binomial Theorem

Expand the expression using the binomial theorem: \[ (y + 1)^4 = \sum_{k=0}^{4} \binom{4}{k} y^{4-k} 1^k \]. This simplifies to: \[ (y + 1)^4 = \binom{4}{0} y^4 1^0 + \binom{4}{1} y^3 1^1 + \binom{4}{2} y^2 1^2 + \binom{4}{3} y^1 1^3 + \binom{4}{4} y^0 1^4 \].

Step 4: Calculate Binomial Coefficients

Calculate each binomial coefficient: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).

Step 5: Substitute Values

Substitute these coefficients back into the expanded formula: \[ (y + 1)^4 = 1 \cdot y^4 \cdot 1 + 4 \cdot y^3 \cdot 1 + 6 \cdot y^2 \cdot 1 + 4 \cdot y^1 \cdot 1 + 1 \cdot y^0 \cdot 1 \].

Step 6: Simplify

Simplify the expression: \[ (y + 1)^4 = y^4 + 4y^3 + 6y^2 + 4y + 1 \].

Key Concepts

Binomial Coefficients

Binomial coefficients are a very important part of the binomial theorem. They are the numbers that appear in Pascal's triangle, and they tell us how to weight each term in the expansion. Each binomial coefficient is written as \(\binom{n}{k}\), where \(n\) is the total number of items, and \(k\) is the number of items chosen. You can calculate a binomial coefficient using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \].
Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\). For instance, in expanding \((y + 1)^4\), the coefficient for each term is found using this formula. Knowing how to find these coefficients is crucial for expanding binomials.

Algebraic Expressions

Algebraic expressions combine numbers and variables using operations like addition, subtraction, multiplication, and division.
In the binomial theorem, we deal with expressions like \(y + 1\). When expanding \((y + 1)^4\), we're creating a new, longer algebraic expression. This expression has multiple terms, each a product of the original terms raised to specific powers and multiplied by a binomial coefficient. Understanding how to manipulate and simplify these expressions helps in many areas of algebra.

Polynomial Expansion

Polynomial expansion transforms a binomial raised to an integer power into a sum of terms. Each term in the polynomial includes a binomial coefficient and powers of the variables involved.
For example, using the binomial theorem to expand \((y + 1)^4\), we get: \[ (y + 1)^4 = y^4 + 4y^3 + 6y^2 + 4y + 1\].
This expanded form is a polynomial where each term results from multiplying the binomial coefficients by the variable \(y\) raised to different powers. This concept helps in understanding the behavior of functions and solving equations.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. It plays a key role in the binomial theorem because it helps us understand how the binomial coefficients are derived.
When expanding \((y + 1)^4\), you're essentially looking at all possible combinations of the terms \(y\) and \(1\) taken 4 at a time.

• The binomial coefficient \( \binom{4}{2} \) tells you the number of ways to pick 2 terms out of 4 as \(y\).
• Each term in the expanded binomial is the result of these combinations.

Understanding combinatorics adds depth to your grasp of the binomial theorem and helps explain why the theorem works the way it does.