%5E4-binomial-expansion.jpeg "(1-2x)^4 Binomial Expansion")
Binomial Expansion of (1-2x)^4
In this article, we will explore the binomial expansion of (1-2x)^4, which is a fundamental concept in algebra and mathematics.
What is Binomial Expansion?
A binomial expansion is the expansion of an expression of the form (a+b)^n, where a and b are constants and n is a positive integer, into a sum of terms involving various powers of a and b. The binomial theorem, which is a mathematical formula, is used to expand such expressions.
Binomial Theorem
The binomial theorem states that:
(a+b)^n = ∑(n choose k) * a^(n-k) * b^k
where n is a positive integer, and k ranges from 0 to n.
Expanding (1-2x)^4
Using the binomial theorem, we can expand (1-2x)^4 as follows:
(1-2x)^4 = ∑(4 choose k) * 1^(4-k) * (-2x)^k
Expanding the summation, we get:
(1-2x)^4 = 1^4 - 4 * 1^3 * (-2x) + 6 * 1^2 * (-2x)^2 - 4 * 1^1 * (-2x)^3 + (-2x)^4
Simplifying the expression, we get:
(1-2x)^4 = 1 - 8x + 24x^2 - 32x^3 + 16x^4
Conclusion
In this article, we have successfully expanded (1-2x)^4 using the binomial theorem. The expansion is a polynomial expression in terms of x, with coefficients obtained using the binomial theorem. This expansion has applications in various areas of mathematics, such as algebra, calculus, and combinatorics.
Reference
- Binomial Theorem:
- Algebra:
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