For the graph of f(x)=x^3+2x^2+x determine the x along with their multiplicities

Answer 1

To determine the x-values and their multiplicities for the graph of f(x) = x^3 + 2x^2 + x, we need to find the roots of the equation f(x) = 0.

Setting f(x) equal to zero, we have:
x^3 + 2x^2 + x = 0

We can factor out common terms to simplify the equation:
x(x^2 + 2x + 1) = 0

The equation x^2 + 2x + 1 can be factored as (x + 1)(x + 1) or (x + 1)^2.

Therefore, the equation can be written as:
x(x + 1)^2 = 0

Setting each factor equal to zero, we have:
x = 0 (multiplicity 1)
x + 1 = 0 (multiplicity 2)

Therefore, the x-values for the graph of f(x) = x^3 + 2x^2 + x, along with their multiplicities, are:
x = 0 (multiplicity 1)
x = -1 (multiplicity 2)