(1+4x)^5 (3+x-x^2)^8

4 min read Jun 16, 2024

Expanding the Binomial Expression (1 + 4x)^5 (3 + x - x^2)^8

This article will explore the expansion of the binomial expression (1 + 4x)^5 (3 + x - x^2)^8. While a full expansion would be quite lengthy, we'll outline the key concepts and methods involved, demonstrating how to approach this problem.

Understanding the Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions of the form (a + b)^n. It states:

(a + b)^n = Σ (n choose k) a^(n-k) b^k

where:

n is a non-negative integer representing the power.
k ranges from 0 to n.
(n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).

Expanding Each Factor Individually

To expand the entire expression, we'll first expand each factor separately using the Binomial Theorem.

Expanding (1 + 4x)^5:

n = 5: The power of the binomial.
a = 1: The first term in the binomial.
b = 4x: The second term in the binomial.

Applying the Binomial Theorem:

(1 + 4x)^5 = (5 choose 0) 1^5 (4x)^0 + (5 choose 1) 1^4 (4x)^1 + (5 choose 2) 1^3 (4x)^2 + (5 choose 3) 1^2 (4x)^3 + (5 choose 4) 1^1 (4x)^4 + (5 choose 5) 1^0 (4x)^5

Simplifying:

(1 + 4x)^5 = 1 + 20x + 160x^2 + 640x^3 + 1024x^4 + 1024x^5

Expanding (3 + x - x^2)^8:

n = 8: The power of the binomial.
a = 3: The first term in the binomial.
b = x - x^2: The second term in the binomial.

Applying the Binomial Theorem:

(3 + x - x^2)^8 = Σ (8 choose k) 3^(8-k) (x - x^2)^k

This expansion would involve numerous terms and require careful calculation of binomial coefficients and powers of (x - x^2).

Combining the Expanded Factors

Once both individual factors are expanded, we can multiply the resulting polynomial expressions together. This involves multiplying each term in the first expansion by every term in the second expansion.

This process would lead to a very large polynomial expression with terms ranging from x^0 to x^13.

Key Points to Note:

The full expansion would be lengthy and complex, but the process outlined provides a structured approach.
The Binomial Theorem is the core tool used to break down the problem.
Using a computer algebra system (CAS) can simplify the calculations and provide the complete expanded expression.

Conclusion

Expanding (1 + 4x)^5 (3 + x - x^2)^8 is a challenging problem that involves applying the Binomial Theorem, performing multiple expansions, and multiplying the resulting polynomials. While the full expansion may be laborious, understanding the process and tools involved is crucial for handling such expressions.