Problem 46 - Algebra for College Students

Question

Discussion. Find the fourth term in the binomial expansion for \((x+y)^{120} .\) Find the fifth term in the binomial expansion for \((x-2 y)^{100} .\) Did you have any trouble computing the coefficients?

Step-by-Step Solution

Answer

The fourth term in \((x + y)^{120}\) is \280840 x^{117} y^3\. The fifth term in \((x - 2y)^{100}\) is \62739600 x^{96} y^4\.

Step 1: Understand the Binomial Expansion

The binomial expansion of \((a + b)^n\) is given by the binomial theorem: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. We will use this formula to find the required terms.

Step 2: Identify the General Term in the Expansion

The general term in the binomial expansion of \((a + b)^n\) is \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \].

Step 3: Find the Fourth Term in \((x + y)^{120}\)

To find the fourth term, set \k = 3\.Using the general term formula: \[ T_4 = \binom{120}{3} x^{120-3} y^3 = \binom{120}{3} x^{117} y^3 \]. Calculate the binomial coefficient: \[ \binom{120}{3} = \frac{120!}{3!(120-3)!} = 280840 \]. Thus, the fourth term is \280840 x^{117} y^3\.

Step 4: Find the Fifth Term in \((x - 2y)^{100}\)

To find the fifth term, set \k = 4\.Using the general term formula: \[ T_5 = \binom{100}{4} x^{100-4} (-2y)^4 = \binom{100}{4} x^{96} (16y^4) \]. Calculate the binomial coefficient: \[ \binom{100}{4} = \frac{100!}{4!(100-4)!} = 3921225 \]. Thus, the fifth term is \3921225 \cdot 16 x^{96} y^4 = 62739600 x^{96} y^4 \.

Step 5: Conclusion

There was no trouble in computing the coefficients using the binomial theorem.

Key Concepts

Binomial Theorem

The binomial theorem helps expand expressions of the form \( (a + b)^n \). It states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]. This equation shows a sum of several terms, each involving a binomial coefficient. This is useful in algebra when you need the terms of a large power expanded rather than writing it all out by multiplying.

Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{k} \), counts the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order of selection. It's calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]. For example, \( \binom{120}{3} \) means calculating \[ \frac{120!}{3!(120-3)!} = 280840 \]. Using these coefficients, we can easily find terms in a binomial expansion.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In the binomial theorem, we see expressions like \(x + y\text \ or \ (x - 2y)\text , \ expanded. For example, in finding the fourth term of \( (x + y)^{120} \), you use algebraic manipulation with binomial coefficients. By substituting the values into the general term formula \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \], like setting \( k = 3 \) for the fourth term in \ (x + y)^{120}, we find \ 280840 x^{117} y^3 \). Algebraic expressions let us apply these rules effectively.