Problem 64 - Algebra and Trigonometry

Question

Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

Step-by-Step Solution

Answer

The expanded form of \((x+y)^3\) is \(x^3 + 3x^2y + 3xy^2 + y^3\)

Step 1: Understanding the Binomial Theorem

The Binomial Theorem is given by: \((a + b)^n = \sum _{k=0} ^n \binom{n}{k} a^{n-k} b^{k}\). Here, \(a\) and \(b\) are the terms of the binomial and \(n\) is the power the binomial is raised to. \(\binom{n}{k}\) are the binomial coefficients, which can be found using the combination formula or Pascal's triangle.

Step 2: Applying the Binomial Theorem for Expansion

Let's see an example by expanding \( (x+y)^3 \). According to the binomial theorem, this is equal to \( \binom{3}{0} x^3 y^0 + \binom{3}{1} x^2 y^1 + \binom{3}{2} x^1 y^2 + \binom{3}{3} x^0 y^3 \)

Step 3: Calculating Binomial Coefficients

To find the coefficients, we can use the combination formula: \(\binom{n}{k}= \frac{n!}{k!(n-k)!}\), where '!' denotes a factorial. In our case, your calculations would be: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\)

Step 4: Applying Binomial Coefficients

Now replace the binomial coefficients in your previous equation. Your final expanded binomial would be \(x^3 + 3x^2y + 3xy^2 + y^3\)

Step 5: Confirm the Solution

Make sure all terms and coefficients align correctly with the rules of the binomial theorem. When you expand a binomial of power \(n\), you should get \(n + 1\) terms in the expansion. Also, each term's power must sum up to \(n\). In our example, we have 3 + 1 = 4 terms and each term's power sums up to 3 which verifies our solution.