(x-3)(x-5)(x-7)-65.jpeg "(x-1)(x-3)(x-5)(x-7)-65")
Exploring the Expression (x-1)(x-3)(x-5)(x-7)-65
This article explores the mathematical expression (x-1)(x-3)(x-5)(x-7)-65 and delves into its properties, factorization, and potential applications.
Understanding the Expression
The expression represents a polynomial of degree 4, meaning it has four terms when expanded. It is composed of four linear factors multiplied together and a constant term.
Expanding the Expression
To understand the expression better, we can expand it by multiplying out the factors:
(x-1)(x-3)(x-5)(x-7)-65 = (x^2 - 4x + 3)(x^2 - 12x + 35) - 65
Further expanding:
(x^2 - 4x + 3)(x^2 - 12x + 35) - 65 = x^4 - 16x^3 + 79x^2 - 148x + 60
Factoring the Expression
Finding the roots of this polynomial (i.e., the values of x that make the expression equal to zero) can help us factor it. However, directly factoring a fourth-degree polynomial can be complex.
One approach is to try substituting specific values for x and observe if the expression becomes zero. We notice that when x=1, 3, 5, or 7, the expression evaluates to zero. This implies that (x-1), (x-3), (x-5), and (x-7) are factors of the polynomial.
Therefore, we can factor the expression as follows:
x^4 - 16x^3 + 79x^2 - 148x + 60 = (x-1)(x-3)(x-5)(x-7)
Since we have already considered the constant term -65, the factorization is complete.
Applications
This type of expression can be used in various contexts, including:
- Polynomial equations: Solving equations involving this expression will require finding its roots.
- Curve fitting: This expression could be used to model certain types of curves or functions.
- Calculus: The derivative and integral of this expression can be used in calculus applications.
Conclusion
While seemingly complex, the expression (x-1)(x-3)(x-5)(x-7)-65 can be understood and factored by recognizing its linear factors. This knowledge can be applied to various mathematical problems and applications involving polynomials.