Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write the expansion of each binomial. $$ (1+y)^{5} $$

Step-by-Step Solution

Answer

The expansion of \((1+y)^5\) is \(1 + 5y + 10y^2 + 10y^3 + 5y^4 + y^5.\)

Step 1: Identify the binomial and the power

The given problem is to expand the binomial \((1 + y)^5\). The binomial consists of two parts, \(1\) and \(y\), and the exponent is \(5\). We will use the Binomial Theorem to expand this expression.

Step 2: Understand the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = 1\) and \(b = y\), with \(n = 5\). We will calculate each term in the expansion.

Step 3: Calculate the binomial coefficients

The binomial coefficient \(\binom{5}{k}\) describes the number of ways to choose \(k\) items from \(5\). For each \(k = 0, 1, 2, 3, 4, 5\), we calculate these as follows:- \(\binom{5}{0} = 1\)- \(\binom{5}{1} = 5\)- \(\binom{5}{2} = 10\)- \(\binom{5}{3} = 10\)- \(\binom{5}{4} = 5\)- \(\binom{5}{5} = 1\).

Step 4: Calculate each term in the expansion

Using the formula from the Binomial Theorem, each term is \(\binom{5}{k} \cdot (1)^{5-k} \cdot y^k\). Thus, the terms are:- For \(k = 0\): \(1 \cdot 1^5 \cdot y^0 = 1\)- For \(k = 1\): \(5 \cdot 1^4 \cdot y^1 = 5y\)- For \(k = 2\): \(10 \cdot 1^3 \cdot y^2 = 10y^2\)- For \(k = 3\): \(10 \cdot 1^2 \cdot y^3 = 10y^3\)- For \(k = 4\): \(5 \cdot 1^1 \cdot y^4 = 5y^4\)- For \(k = 5\): \(1 \cdot 1^0 \cdot y^5 = y^5\).

Step 5: Combine all terms

Adding all terms from Step 4, the expansion is:\((1+y)^5 = 1 + 5y + 10y^2 + 10y^3 + 5y^4 + y^5.\)

Key Concepts

Binomial Expansion

When dealing with expressions such as \((a + b)^n\), the goal is to expand them out completely without actually multiplying everything together directly. This is where the Binomial Expansion comes into play, providing us a methodical way to handle this task. The formula used here is called the Binomial Theorem. It states:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This formula shows how to distribute the powers of a binomial sum into a series of terms. Each term is comprised of parts: a binomial coefficient, and the powers of the individual components \(a\) and \(b\). By using this theorem, we efficiently rewrite these expressions without performing excessive multiplication. This makes it easier to handle larger powers, such as expanding \((1+y)^5\). The coefficients, powers, and the structure are clearly defined, as seen in the expansion result: \(1 + 5y + 10y^2 + 10y^3 + 5y^4 + y^5\).

To become skilled at binomial expansion, practice applying the Binomial Theorem to different binomials, switching up the values of \(a\), \(b\), and \(n\).

Pascal's Triangle

Pascal's Triangle is an intuitive and simple way to find binomial coefficients without performing calculations. By constructing a triangle of numbers, each number in the interior is found by summing the two numbers directly above it, forming a triangular array. This triangle is especially useful in binomial expansion for quickly finding binomial coefficients, as each row in Pascal's Triangle corresponds to the coefficients for the expansion of binomials raised to successive powers.

For instance, in the row corresponding to \(n = 5\), you find the coefficients \(1, 5, 10, 10, 5, 1\), matching exactly with our example's expansion \((1+y)^5\).

• Start with a 1 at the top.
• Continue by adding numbers below, where each number is the sum of the two numbers directly above it.

This technique not only aids in providing coefficients quickly but also provides a deeper connection to patterns found in mathematics, allowing you to apply the Binomial Theorem effectively.

Binomial Coefficients

Binomial coefficients are an integral part of binomial expansion. Represented as \(\binom{n}{k}\), they indicate the number of ways to select \(k\) objects from \(n\) without regard to order. Mathematically, this is expressed as:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

This formula provides a straightforward way to calculate these coefficients. As such, if you are expanding \((a+b)^5\), you end up calculating the coefficients for \(k = 0, 1, 2, 3, 4, 5\). Each coefficient plays a crucial role in terms of its occurrence in the expanded polynomial. The coefficients in our example come out to \(1, 5, 10, 10, 5, 1\), as seen with the expansion of \((1+y)^5\).

Understanding and calculating these coefficients involve comprehension of factorial math and combinatorial principles. It doesn't just stop at solving binomials; these coefficients appear throughout diverse areas like probability theory and statistics, showing their fundamental role in mathematics.