Q 69. Roots of the equation given below are: x^4-4x^3+ 6x^2-4x+1=0 Ops: A. 3,1,3,1 B. 1,1,1,1 C. 1,2,1,2 D. 2,2,2,2

1. Break Down the Problem

We need to find the roots of the polynomial equation given by $x^4 - 4x^3 + 6x^2 - 4x + 1 = 0$.

2. Relevant Concepts

We can use the Binomial Theorem or recognize patterns in the polynomial. Notice that the coefficients correspond to a binomial expansion:

$(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1$

3. Analysis and Detail

Since $(x-1)^4$ simplifies to the given polynomial, it implies that the roots of the polynomial are given by:

$(x-1)^4 = 0$

Solving this gives:

$x - 1 = 0 \implies x = 1$

The root $x = 1$ has a multiplicity of 4.

4. Verify and Summarize

Now we can summarize that the only root is $x = 1$ repeated 4 times.

Final Answer

The roots of the equation are $1, 1, 1, 1$. Therefore, the correct option is B. 1,1,1,1.