Final answer:
The polynomial 1x⁵+5x⁴+10x³+10x²+5x+1 is the expanded form of (x+1)⁵, so written as a product of linear factors, it is (x+1)(x+1)(x+1)(x+1)(x+1).
Step-by-step explanation:
The question asks to write the polynomial 1x⁵+5x⁴+10x³+10x²+5x+1 as a product of linear factors. This polynomial is a symmetric polynomial which suggests that it could be a result of a binomial expansion. In fact, it is the expanded form of (x+1)⁵. To see this, we can recognize it as a binomial expansion using the Pascal's triangle or the binomial theorem.
If we expand (x+1) to the fifth power, we get:
- (x+1)⁵ = x⁵ + 5x⁴ + 10x³ + 10x² + 5x + 1
This matches the original polynomial, so the polynomial as a product of linear factors is:
- (x+1)(x+1)(x+1)(x+1)(x+1) or
- (x+1)⁵