Problem 19 - College Algebra

Question

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(y-3)^{4}$$

Step-by-Step Solution

Answer

The binomial expansion for \((y-3)^4\) is \(y^4 - 12y^3 + 54y^2 - 108y + 81\).

Step 1: Recognize the Variables

In the given expression, identify \(a\), \(b\) and \(n\). Here \(a = y\), \(b = -3\) and \(n = 4\).

Step 2: Apply the Binomial Coefficients

Apply the binomial theorem using the formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). This amounts to 5 terms, one for each success probability from \(k=0\) to \(k=4\).

Step 3: Calculate the Terms and Express in Simplified Form

Calculate each term of the expansion, applying the power to each term and the binomial coefficient:1. Term for \(k=0\): \(\binom{4}{0} y^4 (-3)^0 = 1 * y^4 * 1 = y^4\)2. Term for \(k=1\): \(\binom{4}{1} y^3 (-3)^1 = 4 * y^3 * -3 = -12y^3\)3. Term for \(k=2\): \(\binom{4}{2} y^2 (-3)^2 = 6 * y^2 * 9 = 54y^2\)4. Term for \(k=3\): \(\binom{4}{3} y^1 (-3)^3 = 4 * y * -27 = -108y\)5. Term for \(k=4\): \(\binom{4}{4} y^0 (-3)^4 = 1 * 1 * 81 = 81\)

Step 4: Write the Final Simplified Expansion

Combine all terms calculated in step 3 to formulate the final simplified expansion of the binomial: \(y^4 - 12y^3 + 54y^2 - 108y + 81\).

Key Concepts

Binomial Coefficients

When we talk about binomial coefficients, we're referring to the specific numbers that are vital in the expansion of binomials according to the Binomial Theorem. These coefficients appear in the series famously known as Pascal's triangle. In mathematical notation, a binomial coefficient is represented as \(\binom{n}{k}\), where \(n\) is the power to which the binomial is raised, and \(k\) is the specific term's position in the expansion, starting at 0.

For a given integer \(n\) and \(k\), the binomial coefficient \(\binom{n}{k}\) is calculated using the factorial function: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers up to \(n\). This is a systematic method to determine how many ways you can choose \(k\) items from a set of \(n\) distinct items.

In the context of our textbook example, \(\binom{4}{k}\) will calculate the binomial coefficients for each term when expanding \( (y-3)^4 \). These coefficients dictate the relative weight of each term in the polynomial expansion.

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial, \( (a+b)^n \), as the sum of terms involving different powers of \( a \) and \( b \) multiplied by the binomial coefficients. The Binomial Theorem provides us with a formula for this expansion: \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). This formula tells us each term in the expansion is a combination of \( a \) and \( b \) raised to complementing powers that always sum up to \( n \).

The wonderful thing about this theorem is that it applies universally to any binomial and any positive integer exponent. As demonstrated in the solution for \( (y-3)^4 \) in our textbook, the result is a polynomial where each term comes from the application of the theorem, generating an algebraic expression which precisely represents the expanded form of the original binomial power.

Simplified Form Expression

A simplified form expression in the context of binomial expansion is the written out form of the polynomial that results from applying the Binomial Theorem, where all like terms have been combined and the powers of the terms are in descending order. This process not only involves calculating each term using binomial coefficients but also simplifying the expression to make it compact and easily understandable.

In the step-by-step solution as shown, after applying the binomial coefficients to calculate the value of each term in the expansion of \( (y-3)^4 \), the polynomial is simplified by multiplying and adding together each term in the sequence. The final simplified expression for our example is \( y^4 - 12y^3 + 54y^2 - 108y + 81 \), with the understanding that there's no further simplification possible since there are no like terms to combine. This process is crucial for presenting the expanded polynomial in a form that is ready for further mathematical exploration or application.