(x%5E2-3x-5).jpeg "(x^2-2x-4)(x^2-3x-5)")
Expanding the Expression: (x^2 - 2x - 4)(x^2 - 3x - 5)
This expression represents the product of two quadratic expressions. To simplify it, we need to expand the product using the distributive property (also known as FOIL).
Step 1: Distribute the first term of the first expression.
(x^2 - 2x - 4)(x^2 - 3x - 5) = x^2(x^2 - 3x - 5) - 2x(x^2 - 3x - 5) - 4(x^2 - 3x - 5)
Step 2: Distribute further.
= x^4 - 3x^3 - 5x^2 - 2x^3 + 6x^2 + 10x - 4x^2 + 12x + 20
Step 3: Combine like terms.
= x^4 - 5x^3 - 3x^2 + 22x + 20
Therefore, the expanded form of the expression (x^2 - 2x - 4)(x^2 - 3x - 5) is x^4 - 5x^3 - 3x^2 + 22x + 20.
Further Analysis:
This expanded expression represents a quartic polynomial (a polynomial with the highest power of the variable being 4). It can be factored further, but it might require more advanced techniques or the use of the quadratic formula.
Applications:
Understanding how to expand expressions like this is crucial in various areas of mathematics, including:
- Algebra: Solving equations, simplifying expressions, and working with polynomials.
- Calculus: Finding derivatives and integrals of functions.
- Physics and Engineering: Modeling real-world phenomena.