(x%5E4%2Bx%5E3%2Bx%5E2%2Bx%2B1).jpeg "(x-1)(x^4+x^3+x^2+x+1)")
Expanding (x-1)(x⁴+x³+x²+x+1)
This expression represents a product of two polynomials: a linear term (x-1) and a polynomial of degree four (x⁴+x³+x²+x+1). To understand the expression better, let's expand it using the distributive property.
Applying the Distributive Property
We multiply each term of the linear polynomial by each term of the quartic polynomial:
- (x) (x⁴+x³+x²+x+1) = x⁵ + x⁴ + x³ + x² + x
- (-1) (x⁴+x³+x²+x+1) = -x⁴ - x³ - x² - x - 1
Combining Terms
Now, we add the results from the above steps and combine like terms:
x⁵ + x⁴ + x³ + x² + x -x⁴ - x³ - x² - x - 1
This simplifies to x⁵ - 1.
Conclusion
The expanded form of (x-1)(x⁴+x³+x²+x+1) is x⁵ - 1. This highlights an interesting pattern: the product of a linear term of the form (x-a) and a polynomial with terms that are consecutive powers of x, from x⁴ down to x⁰, results in a difference of powers: x⁵ - a⁵. This pattern applies generally, where the exponent of x in the final result is one more than the highest power in the original polynomial.