Problem 24 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(y - 2\right)^5 \)

Step-by-Step Solution

Answer

The expanded form of the expression \( (y - 2)^5 \) using the binomial theorem is \( y^5 - 10y^4*2 + 40y^3*2^2 - 80y^2*2^3 + 80y*2^4 - 2^5 \), which simplifies to \( y^5 - 10y^4*2 + 40y^3*4 - 80y^2*8 + 80y*16 - 32 \), and then to \( y^5 - 20y^4 + 160y^3 - 640y^2 + 1280y - 32 \).

Step 1: Understand the Binomial Theorem

The formula for binomial expansion, or Binomial Theorem, for any real numbers a and b, and natural number n, is defined as: \[ (a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{n-1}ab^{n-1} + b^n \] Where \( \binom{n}{r} \) is number of ways to choose r from n, also known as 'n choose r' or binomial coefficient. It equals \( \frac{n!}{r!(n-r)!} \). n! represents the factorial of the number n, which is the product of all positive integers up to n.

Step 2: Apply the Binomial Theorem

Now, apply binomial theorem to expand \( (y - 2)^5 \). Note that in this case, a is y, b is -2 and n is 5.

Step 3: Calculate the Terms

Start calculating terms one by one. Also remember that negative powers can be written as \( b^r = (-2)^r \) and it might be positive or negative depending on whether r is even or odd.

Step 4: Sum up all Terms

Finally sum up all the terms, and simplify the expression. You should get the expanded form of the expression.

Key Concepts

Binomial expansion

The Binomial Theorem allows us to expand expressions of the form \( (a + b)^n \) into a sum of terms with a specific pattern. This pattern is known as binomial expansion. Binomial expansion helps break down complex expressions into simpler components, which is particularly useful in algebra and calculus. For example, if you have an expression such as \( (y - 2)^5 \), you can express it as a series of terms using the binomial theorem.The theorem shows that \( (a + b)^n \) is equal to the sum of terms formed by binomial coefficients, powers of \( a \), and powers of \( b \). Each term in the expansion is calculated by choosing how many times you want \( b \) rather than \( a \). When you apply the theorem, it provides a straightforward way to compute each term sequentially, leading to an expanded expression with all the potential integer powers.Understanding binomial expansion is key for handling polynomial expressions and is foundational in many areas of mathematics. By breaking down a problem into manageable parts, you are simplifying your work and making it easier to solve.

Binomial coefficient

The binomial coefficient, often referred to with the notation \( \binom{n}{r} \), plays a crucial role in the process of binomial expansion. This coefficient tells us the number of ways we can choose \( r \) elements from a set of \( n \) elements, and it's an integral part of calculating each term in a binomial expansion.Mathematically, binomial coefficients are calculated using the formula:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where \( n! \) (read as \"n factorial\") is the factorial of \( n \). This computation is essential in generating the correct coefficient for each term in the binomial series. Understanding how to compute binomial coefficients is vital for efficiently expanding binomial expressions without manually expanding each term.In the example of \( (y - 2)^5 \), you would calculate each term's coefficient by determining \( \binom{5}{r} \) for each power \( r \) from 0 to 5, ensuring the proper proportions between \( y \) and \( -2 \) in each term. These coefficients facilitate the expansion process, saving significant computational time and effort.

Factorial

Factorials are an important mathematical concept, especially when dealing with binomial coefficients. A factorial, represented as \( n! \), means the product of all positive integers from 1 to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow rapidly with increasing \( n \), which makes them integral in various combinatorial and statistical calculations.To understand factorials in the context of the binomial theorem, consider the formula for the binomial coefficient \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, factorials are used to calculate how many unique ways we can arrange \( n \) objects by extracting \( r \) parts. Factorials ensure that the order and difference between combinations are properly accounted for.When applying the binomial theorem, knowing how to compute and utilize factorials securely will enhance your efficiency in solving problems. In expanding \( (y - 2)^5 \), you repeat calculations involving factorials for each term to find correct coefficients. Thoroughly understanding factorials will make complex expressions seem far less daunting.